problem solving on division of polynomials

Division of Polynomials – Example and Practice Problems

Polynomials can be divided using long division of polynomials. Dividing the polynomials in this format allows us to better visualize each of the steps involved. If we get a remainder after doing the division, we must include it in the final answer by writing it as a fraction.

Here, we will look at a summary of the process used to divide polynomials. Also, we will explore several examples with answers of the division of polynomials in order to visualize the application of this process.

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Exploring examples of division of polynomials.

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Summary of division of polynomials

Division of polynomials – examples with answers, division of polynomials – practice problems.

To divide polynomials that contain more than one term, we have to use the so-called long division of polynomials. We carry out the long division of polynomials by following these steps:

Step 1: We have to make sure that the polynomial is written in descending order. If there are any missing terms, we use a zero to fill a space or we just leave a blank space.

Step 2: We divide the term with the greatest power inside the division symbol by the term with the greatest power outside the division symbol.

Step 3: We multiply or distribute the answer obtained in the previous step by the polynomial in front of the division symbol.

Step 4: We subtract the obtained expression and write the next term.

Step 5: We repeat steps 2, 3, and 4 until there are no more terms remaining.

Step 6: We write the final answer. The remaining term after the last terms have been subtracted is the remainder. We must write the remainder as a fraction in the final answer.

The long division process mentioned above is used to solve the following polynomial division examples. It is recommended that you try to solve the exercises yourself before looking at the solution.

Solve the division of polynomials: $latex \frac{{{x}^2}+8x+15}{x+5}$.

Step 1: The polynomials are already arranged in descending order.

Step 2: We start dividing the $latex {{x}^2}$ by x , which is equal to x .

Step 3: By multiplying this answer by the polynomial in front $latex (x + 5)$, we have $latex {{x}^2}+5x$.

Step 4: We subtract this expression and get $latex 3x$. We place down the 15 to complete the polynomial.

Step 5: When dividing $latex 3x$ by x , we get 3. We multiply 3 by $latex x+5$ to get $latex 3x+15$. When subtracting we get zero.

Step 6: The final answer is $latex x+3$.

What is the result of this division?: $latex \frac{2{{x}^3}+7{{x}^2}+10x+8}{x+2}$.

Step 1: Here, the polynomials are also arranged in descending order.

Step 2: We start dividing the $latex 2{{x}^3}$ by x , which is equal to $latex 2{{x}^2}$ .

Step 3: We multiply this by the polynomial $latex x+2$, to get $latex 2{{x}^3}+4{{x}^2}$.

Step 4: We subtract this expression to get $latex 3{{x}^2}$. We place down the 10 x to complete the polynomial.

Step 5: When dividing $latex 3{{x}^2}$ by x , we get 3x. Multiplying and subtracting, we have $latex 4x$. We place down the 8 to form $latex 4x+8$. By dividing $latex 4x$ by x , we have 4. Multiplying and subtracting, we have 0.

Step 6: The final answer is $latex 2{{x}^2}+3x+4$.

Solve the division: $latex \frac{{{x}^2}-3x+6}{x+2}$.

Step 1: The polynomials are already organized.

Step 2: We start by dividing the $latex {{x}^2}$ by x , to get $latex x$ .

Step 3: Multiplying by $latex x+2$, we get $latex {{x}^2}+4x$.

Step 4: Subtracting, we get $latex -5x$. We place down the 6 to complete the polynomial.

Step 5: By dividing $latex -5x$ by x , we have -5. Multiplying and subtracting, we have 16.

Step 6: The final answer is $latex x-5+\frac{16}{x+2}$.

What is the result of the division?: $latex \frac{{{x}^5}+{{x}^2}+{{x}^4}+{{x}^3}+x+1}{x-1}$.

Step 1: We start by organizing the polynomials in descending order.

Step 2: We divide the $latex {{x}^5}$ by x , to get $latex {{x}^4}$ .

Step 3: Multiplying this by $latex x-1$, we get $latex {{x}^5}-{{x}^4}$.

Step 4: Subtracting this expression, we have $latex 2{{x}^4}$. We place down to the $latex{{x}^3}$ to complete the polynomial.

Step 5: We repeat steps 2, 3, and 4 until completing the division and obtaining the 6 as the remainder.

Step 6: The final answer is $latex {{x}^4}+2{{x}^3}+\frac{6}{x-1}$.

Solve the division $latex \frac{{{x}^8}+{{x}^7}+{{x}^4}+{{x}^5}+{{x}^6}+{{x}^3}+{{x}^2}+x}{x+1}$.

Step 1: We have to order the polynomials in descending order and leave space if there are not all the terms.

Step 2: We start by dividing the $latex {{x}^8}$ by x , to get $latex {{x}^7}$ .

Step 3: We multiply this by the polynomial $latex x+1$, to get $latex {{x}^8}+{{x}^7}$.

Step 4: We subtract this expression to get 0. We place down the next terms to divide.

Step 5: By dividing $latex {{x}^6}$ by x , we get $latex {{x}^5}$. Multiplying and subtracting, we have 0.

Step 6: We can observe a pattern in the division. The terms always cancel leaving a remainder of 0. Since we have an even number of terms in the divisor, the remainder will be 0 and the quotient will be $latex {{x}^7}+{{x}^5}+{{x}^3}+x$.

→ Polynomial Division Calculator

Practice dividing polynomials using the following problems. Solve the exercises and select the answer obtained. Click “Check” to verify that you got the correct answer.

Divide the polynomials $latex \frac{12{{x}^3}-11{{x}^2}+9x+18}{4x+3}$.

Choose an answer

Divide the polynomials $latex \frac{4{{x}^3}-2{{x}^2}-3}{2{{x}^2}-1}$.

Divide the polynomials $latex \frac{2{{x}^3}+4{{x}^2}-5}{x+3}$., divide the polynomials $latex \frac{{{x}^3}-4{{x}^2}+2x+5}{x-2}$.

Interested in learning more about operations with polynomials? Take a look at these pages:

Polynomial Division Calculator
Examples of Addition of Polynomials
Examples of Subtraction of Polynomials
Examples of Multiplication of Monomials
Examples of Multiplication of Polynomials

Jefferson Huera Guzman

Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.

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Long Division Polynomial

Polynomial long division.

In this lesson, I will go over five (5) examples with detailed step-by-step solutions on how to divide polynomials using the long division method . It is very similar to what you did back in elementary when you try to divide large numbers, for instance, you have [latex]1,723 \div 5[/latex]. You would solve it just like below, right?

Quick Review of the Long Division Method of Numbers

If you can do the simple numerical division by the long method, as shown above, I am convinced that you can do the problems below. The only added thing is the division of variables.

Examples of Dividing Polynomials using the Long Division Method

Example 1 : Divide using the long division method

Solution : I need to make sure that both the dividend (stuff being divided) and the divisor are in the standard form. A polynomial in standard form guarantees that its exponents are in decreasing order from left to right. Performing a quick check on this helps us to prevent basic avoidable mistakes later on.

By quick examination, I hope you agree that both our dividend and divisor are indeed in standard form. That means we are now ready to perform the procedure.

STEP 1 : Consider both the leading terms of the dividend and divisor.

STEP 2 : Divide the leading term of the dividend by the leading term of the divisor.

STEP 3 : Place the partial quotient on top.

STEP 4 : Now take the partial quotient you placed on top, [latex]3x[/latex], and distribute it into the divisor [latex]\left( {2x + 4} \right)[/latex].

STEP 5 : Position the product of [latex]\left( {3x} \right)[/latex] and [latex]\left( {2x + 4} \right)[/latex] under the dividend. Make sure to align them by similar terms.

STEP 6 : Perform subtraction by switching the signs of the bottom polynomial.

STEP 7 : Proceed with regular addition vertically. Notice that the first column from the left cancels each other out. Nice!

STEP 8 : Carry down the next adjacent “unused” term of the dividend.

STEP 9 : Next, look at the bottom polynomial, [latex] – 14x – 28[/latex], take its leading term which is [latex] – 14x[/latex] and divide it by the leading term of the divisor, [latex]2x[/latex].

STEP 10 : Again, place the partial quotient on top.

STEP 11 : Use the partial quotient that you put up, [latex]-7[/latex], and distribute it into the divisor. Seeing a pattern now?

STEP 12 : Place the product of [latex]-7[/latex] and the divisor below as the last line of polynomial entry.

STEP 13 : Subtraction means you will switch the signs (in red).

STEP 14 : Perform regular addition along the columns of similar terms

STEP 15 : This is great because the remainder is zero. It means the divisor is a factor of the dividend.

The final answer is just the stuff on top of the division symbol.

Example 2 : Divide using the long division method

Solution : This problem is also considered “nice” just like the first one because both the dividend and divisor are in standard forms.

This time around you are dividing a polynomial with four terms by a binomial . Remember that example 1 is a division of polynomial with three terms (trinomial) by a binomial. Hopefully, you see a slight difference.

Let’s go ahead and work this out!

STEP 1 : Focus on the leftmost terms of both the dividend and divisor.

STEP 2 : Divide the leftmost term of the dividend by the leftmost term of the divisor.

STEP 3 : Place the partial answer on top.

STEP 4 : Use that partial answer, [latex]{x^2}[/latex], to multiply into the divisor [latex]\left( {3x – 2} \right)[/latex].

STEP 5 : Place their product under the dividend. Make sure to align them by similar terms.

STEP 6 : Perform subtraction by alternating the signs of the bottom polynomial.

STEP 7 : Proceed with regular addition vertically. Again the first column cancels each other out. Looks like a pattern to me!

STEP 8 : Carry down the next adjacent “unused” term of the dividend

STEP 9 : Take the leftmost term of the bottom polynomial, and divide by the leftmost term of the divisor.

STEP 10 : Place the answer on top, as usual.

STEP 11 : Okay, perform another multiplication by the partial answer [latex]2x[/latex] and divisor [latex]\left( {3x – 2} \right)[/latex]. Bring the product below.

STEP 12 : Perform subtraction by switching signs and proceed with normal addition.

STEP 13 : Carry down the last unused term of the dividend. We’re almost there!

STEP 14 : We are going up one more time. Divide the leading term of the bottom polynomial by the leading term of the divisor. Place the answer up there!

STEP 15 : This is our “last trip” going down so we distribute the partial answer [latex]−1[/latex] by the divisor [latex]\left( {3x – 2} \right)[/latex], and place the product “downstairs”.

STEP 16 : Finish this off by subtraction leaving as with a remainder of [latex]-7[/latex].

STEP 17 : Write the final answer in the following form.

Example 3 : Divide using the long division method

Solution : If you observe the dividend, it is missing some powers of variable [latex]x[/latex] which are [latex]{x^3}[/latex] and [latex]{x^2}[/latex]. I need to insert zero coefficients as placeholders for the missing powers of the variable. It is a critical part to correctly apply the procedures in long division.

So I rewrite the original problem as

Now all [latex]x[/latex]’s are accounted for!

STEP 1 : Focus on the leading terms inside and outside the division symbol.

STEP 2 : Divide the first term of the dividend by the first term of the divisor.

STEP 3 : Position the partial answer on top.

STEP 4 : Use that partial answer placed on top, [latex]3{x^2}[/latex] to distribute into the divisor [latex]\left( {x + 1} \right)[/latex].

STEP 5 : Put the result under the dividend. Make sure to align them by similar terms.

STEP 6 : Subtract them together by making sure to switch the signs of the bottom terms before adding.

STEP 7 : Carry down the next unused term of the dividend.

STEP 8 : Looking at the bottom polynomial, [latex] – 3{x^3} + 0{x^2}[/latex], use the leading term [latex] – 3{x^3}[/latex] and divide it by the leading term of the divisor, [latex]x[/latex]. Put the answer above the division symbol.

STEP 9 : Multiply the answer you got previously, [latex] – 3{x^3}[/latex], and distribute into the divisor [latex]\left( {x + 1} \right)[/latex].

STEP 10 : Place the answer below then perform subtraction.

STEP 11 : Bring down the next adjacent term of the dividend

STEP 12 : Go up again by dividing the leading term below by the leading term of the divisor.

STEP 13 : Go down by distributing the answer in partial quotient into the divisor, followed by subtraction.

I believe the pattern makes sense now. Yes?

STEP 14 : Carry down the last term of the dividend.

STEP 15 : Go up again while performing division.

STEP 16 : Go down again while performing multiplication.

STEP 17 : Do the final subtraction, and we are done! The remainder is equal to 20.

STEP 18 : The final answer is

Example 4 : Divide the given polynomial using the long division method

Solution : The dividend is obviously missing a lot of variable [latex]x[/latex]. That means I need to insert zero coefficients in every missing power of the variable.

I need to rewrite the problem this way to include all exponents of [latex]x[/latex] in descending order:

Remember the Main Steps in Long Division:

When going up, we divide
When going down, we distribute
Repeat the process until done

Verify if the steps are being applied correctly in the example below.

So the final answer is

Example 5 : Divide the given polynomial using the long division method

Solution : We have a polynomial with five terms being divided by a trinomial. Both the dividend and divisor are in standard form, and all powers of the variable [latex]x[/latex] are present. This is wonderful because we can now start solving it.

The solution to this problem is presented in the animated picture. Observe each step carefully, and see if you can follow it.

5.4 Dividing Polynomials

Learning objectives.

Dividing monomials
Dividing a polynomial by a monomial
Dividing polynomials using long division
Dividing polynomials using synthetic division
Dividing polynomial functions
Use the remainder and factor theorems

Be Prepared 5.4

Before you get started, take this readiness quiz.

Add: 3 d + x d . 3 d + x d . If you missed this problem, review Example 1.28 .
Simplify: 30 x y 3 5 x y . 30 x y 3 5 x y . If you missed this problem, review Example 1.25 .
Combine like terms: 8 a 2 + 12 a + 1 + 3 a 2 − 5 a + 4 . 8 a 2 + 12 a + 1 + 3 a 2 − 5 a + 4 . If you missed this problem, review Example 1.7 .

Dividing Monomials

We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and polynomials.

Example 5.36

Find the quotient: 54 a 2 b 3 ÷ ( −6 a b 5 ) . 54 a 2 b 3 ÷ ( −6 a b 5 ) .

When we divide monomials with more than one variable, we write one fraction for each variable. 54 a 2 b 3 ÷ ( −6 a b 5 ) Rewrite as a fraction. 54 a 2 b 3 −6 a b 5 Use fraction multiplication. 54 −6 · a 2 a · b 3 b 5 Simplify and use the Quotient Property. −9 · a · 1 b 2 Multiply. − 9 a b 2 54 a 2 b 3 ÷ ( −6 a b 5 ) Rewrite as a fraction. 54 a 2 b 3 −6 a b 5 Use fraction multiplication. 54 −6 · a 2 a · b 3 b 5 Simplify and use the Quotient Property. −9 · a · 1 b 2 Multiply. − 9 a b 2

Try It 5.71

Find the quotient: −72 a 7 b 3 ÷ ( 8 a 12 b 4 ) . −72 a 7 b 3 ÷ ( 8 a 12 b 4 ) .

Try It 5.72

Find the quotient: −63 c 8 d 3 ÷ ( 7 c 12 d 2 ) . −63 c 8 d 3 ÷ ( 7 c 12 d 2 ) .

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Example 5.37

Find the quotient: 14 x 7 y 12 21 x 11 y 6 . 14 x 7 y 12 21 x 11 y 6 .

Be very careful to simplify 14 21 14 21 by dividing out a common factor, and to simplify the variables by subtracting their exponents. 14 x 7 y 12 21 x 11 y 6 Simplify and use the Quotient Property. 2 y 6 3 x 4 14 x 7 y 12 21 x 11 y 6 Simplify and use the Quotient Property. 2 y 6 3 x 4

Try It 5.73

Find the quotient: 28 x 5 y 14 49 x 9 y 12 . 28 x 5 y 14 49 x 9 y 12 .

Try It 5.74

Find the quotient: 30 m 5 n 11 48 m 10 n 14 . 30 m 5 n 11 48 m 10 n 14 .

Divide a Polynomial by a Monomial

Now that we know how to divide a monomial by a monomial, the next procedure is to divide a polynomial of two or more terms by a monomial.

The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition. The sum y 5 + 2 5 y 5 + 2 5 simplifies to y + 2 5 . y + 2 5 .

Now we will do this in reverse to split a single fraction into separate fractions. For example, y + 2 5 y + 2 5 can be written y 5 + 2 5 . y 5 + 2 5 .

This is the “reverse” of fraction addition and it states that if a , b , and c are numbers where c ≠ 0 , c ≠ 0 , then a + b c = a c + b c . a + b c = a c + b c . We will use this to divide polynomials by monomials.

Division of a Polynomial by a Monomial

To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.

Example 5.38

Find the quotient: ( 18 x 3 y − 36 x y 2 ) ÷ ( −3 x y ) . ( 18 x 3 y − 36 x y 2 ) ÷ ( −3 x y ) .

( 18 x 3 y − 36 x y 2 ) ÷ ( −3 x y ) Rewrite as a fraction. 18 x 3 y − 36 x y 2 −3 x y Divide each term by the divisor. Be careful with the signs! 18 x 3 y −3 x y − 36 x y 2 −3 x y Simplify. −6 x 2 + 12 y ( 18 x 3 y − 36 x y 2 ) ÷ ( −3 x y ) Rewrite as a fraction. 18 x 3 y − 36 x y 2 −3 x y Divide each term by the divisor. Be careful with the signs! 18 x 3 y −3 x y − 36 x y 2 −3 x y Simplify. −6 x 2 + 12 y

Try It 5.75

Find the quotient: ( 32 a 2 b − 16 a b 2 ) ÷ ( −8 a b ) . ( 32 a 2 b − 16 a b 2 ) ÷ ( −8 a b ) .

Try It 5.76

Find the quotient: ( −48 a 8 b 4 − 36 a 6 b 5 ) ÷ ( −6 a 3 b 3 ) . ( −48 a 8 b 4 − 36 a 6 b 5 ) ÷ ( −6 a 3 b 3 ) .

Divide Polynomials Using Long Division

Divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

We check division by multiplying the quotient by the divisor.

If we did the division correctly, the product should equal the dividend.

Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.

Example 5.39

Find the quotient: ( x 2 + 9 x + 20 ) ÷ ( x + 5 ) . ( x 2 + 9 x + 20 ) ÷ ( x + 5 ) .


Write it as a long division problem. Be sure the dividend is in standard form.
Divide by It may help to ask yourself, “What do I need to multiply by to get ?”
Put the answer, in the quotient over the term. Multiply times Line up the like terms under the dividend.
Subtract from You may find it easier to change the signs and then add. Then bring down the last term, 20.
Divide by It may help to ask yourself, “What do I need to multiply by to get ?” Put the answer, , in the quotient over the constant term.
Multiply 4 times
Subtract from
Check: Multiply the quotient by the divisor. You should get the dividend.

Try It 5.77

Find the quotient: ( y 2 + 10 y + 21 ) ÷ ( y + 3 ) . ( y 2 + 10 y + 21 ) ÷ ( y + 3 ) .

Try It 5.78

Find the quotient: ( m 2 + 9 m + 20 ) ÷ ( m + 4 ) . ( m 2 + 9 m + 20 ) ÷ ( m + 4 ) .

When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In the next example, we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.

Look back at the dividends in previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be x 4 − x 2 + 5 x − 6 . x 4 − x 2 + 5 x − 6 . It is missing an x 3 x 3 term. We will add in 0 x 3 0 x 3 as a placeholder.

Example 5.40

Find the quotient: ( x 4 − x 2 + 5 x − 6 ) ÷ ( x + 2 ) . ( x 4 − x 2 + 5 x − 6 ) ÷ ( x + 2 ) .

Notice that there is no x 3 x 3 term in the dividend. We will add 0 x 3 0 x 3 as a placeholder.


Write it as a long division problem. Be sure the dividend is in standard form with placeholders for missing terms.
Divide by Put the answer, in the quotient over the term. Multiply times Line up the like terms. Subtract and then bring down the next term.
Divide by Put the answer, in the quotient over the term. Multiply times Line up the like terms Subtract and bring down the next term.
Divide by Put the answer, in the quotient over the term. Multiply times Line up the like terms. Subtract and bring down the next term.
Divide by Put the answer, in the quotient over the constant term. Multiply times Line up the like terms. Change the signs, add. Write the remainder as a fraction with the divisor as the denominator.
To check, multiply . The result should be

Try It 5.79

Find the quotient: ( x 4 − 7 x 2 + 7 x + 6 ) ÷ ( x + 3 ) . ( x 4 − 7 x 2 + 7 x + 6 ) ÷ ( x + 3 ) .

Try It 5.80

Find the quotient: ( x 4 − 11 x 2 − 7 x − 6 ) ÷ ( x + 3 ) . ( x 4 − 11 x 2 − 7 x − 6 ) ÷ ( x + 3 ) .

In the next example, we will divide by 2 a + 3 . 2 a + 3 . As we divide, we will have to consider the constants as well as the variables.

Example 5.41

Find the quotient: ( 8 a 3 + 27 ) ÷ ( 2 a + 3 ) . ( 8 a 3 + 27 ) ÷ ( 2 a + 3 ) .

This time we will show the division all in one step. We need to add two placeholders in order to divide.

To check, multiply ( 2 a + 3 ) ( 4 a 2 − 6 a + 9 ) . ( 2 a + 3 ) ( 4 a 2 − 6 a + 9 ) .

The result should be 8 a 3 + 27 . 8 a 3 + 27 .

Try It 5.81

Find the quotient: ( x 3 − 64 ) ÷ ( x − 4 ) . ( x 3 − 64 ) ÷ ( x − 4 ) .

Try It 5.82

Find the quotient: ( 125 x 3 − 8 ) ÷ ( 5 x − 2 ) . ( 125 x 3 − 8 ) ÷ ( 5 x − 2 ) .

Divide Polynomials using Synthetic Division

As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in Example 5.39 and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.

Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the x x and x 2 x 2 are removed. as well as the − x 2 − x 2 and −4 x −4 x as they are opposite the term above.

The first row of the synthetic division is the coefficients of the dividend. The −5 −5 is the opposite of the 5 in the divisor.

The second row of the synthetic division are the numbers shown in red in the division problem.

The third row of the synthetic division are the numbers shown in blue in the division problem.

Notice the quotient and remainder are shown in the third row.

The following example will explain the process.

Example 5.42

Use synthetic division to find the quotient and remainder when 2 x 3 + 3 x 2 + x + 8 2 x 3 + 3 x 2 + x + 8 is divided by x + 2 . x + 2 .

Write the dividend with decreasing powers of
Write the coefficients of the terms as the first row of the synthetic division.
Write the divisor as and place in the synthetic division in the divisor box.
Bring down the first coefficient to the third row.
Multiply that coefficient by the divisor and place the result in the second row under the second coefficient.
Add the second column, putting the result in the third row.
Multiply that result by the divisor and place the result in the second row under the third coefficient.
Add the third column, putting the result in the third row.
Multiply that result by the divisor and place the result in the third row under the third coefficient.
Add the final column, putting the result in the third row.
The quotient is and the remainder is 2.

The division is complete. The numbers in the third row give us the result. The 2 −1 3 2 −1 3 are the coefficients of the quotient. The quotient is 2 x 2 − 1 x + 3 . 2 x 2 − 1 x + 3 . The 2 in the box in the third row is the remainder.

(quotient)(divisor) + remainder = dividend ( 2 x 2 − 1 x + 3 ) ( x + 2 ) + 2 = ? 2 x 3 + 3 x 2 + x + 8 2 x 3 − x 2 + 3 x + 4 x 2 − 2 x + 6 + 2 = ? 2 x 3 + 3 x 2 + x + 8 2 x 3 + 3 x 2 + x + 8 = 2 x 3 + 3 x 2 + x + 8 ✓ (quotient)(divisor) + remainder = dividend ( 2 x 2 − 1 x + 3 ) ( x + 2 ) + 2 = ? 2 x 3 + 3 x 2 + x + 8 2 x 3 − x 2 + 3 x + 4 x 2 − 2 x + 6 + 2 = ? 2 x 3 + 3 x 2 + x + 8 2 x 3 + 3 x 2 + x + 8 = 2 x 3 + 3 x 2 + x + 8 ✓

Try It 5.83

Use synthetic division to find the quotient and remainder when 3 x 3 + 10 x 2 + 6 x − 2 3 x 3 + 10 x 2 + 6 x − 2 is divided by x + 2 . x + 2 .

Try It 5.84

Use synthetic division to find the quotient and remainder when 4 x 3 + 5 x 2 − 5 x + 3 4 x 3 + 5 x 2 − 5 x + 3 is divided by x + 2 . x + 2 .

In the next example, we will do all the steps together.

Example 5.43

Use synthetic division to find the quotient and remainder when x 4 − 16 x 2 + 3 x + 12 x 4 − 16 x 2 + 3 x + 12 is divided by x + 4 . x + 4 .

The polynomial x 4 − 16 x 2 + 3 x + 12 x 4 − 16 x 2 + 3 x + 12 has its term in order with descending degree but we notice there is no x 3 x 3 term. We will add a 0 as a placeholder for the x 3 x 3 term. In x − c x − c form, the divisor is x − ( −4 ) . x − ( −4 ) .

We divided a 4 th degree polynomial by a 1 st degree polynomial so the quotient will be a 3 rd degree polynomial.

Reading from the third row, the quotient has the coefficients 1 −4 0 3 , 1 −4 0 3 , which is x 3 − 4 x 2 + 3 . x 3 − 4 x 2 + 3 . The remainder is 0.

Try It 5.85

Use synthetic division to find the quotient and remainder when x 4 − 16 x 2 + 5 x + 20 x 4 − 16 x 2 + 5 x + 20 is divided by x + 4 . x + 4 .

Try It 5.86

Use synthetic division to find the quotient and remainder when x 4 − 9 x 2 + 2 x + 6 x 4 − 9 x 2 + 2 x + 6 is divided by x + 3 . x + 3 .

Divide Polynomial Functions

Just as polynomials can be divided, polynomial functions can also be divided.

Division of Polynomial Functions

For functions f ( x ) f ( x ) and g ( x ) , g ( x ) , where g ( x ) ≠ 0 , g ( x ) ≠ 0 ,

Example 5.44

For functions f ( x ) = x 2 − 5 x − 14 f ( x ) = x 2 − 5 x − 14 and g ( x ) = x + 2 , g ( x ) = x + 2 , find: ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( −4 ) . ( f g ) ( −4 ) .

Substitute for f ( x ) and g ( x ) . ( f g ) ( x ) = x 2 − 5 x − 14 x + 2 Divide the polynomials. ( f g ) ( x ) = x − 7 Substitute for f ( x ) and g ( x ) . ( f g ) ( x ) = x 2 − 5 x − 14 x + 2 Divide the polynomials. ( f g ) ( x ) = x − 7 ⓑ In part ⓐ we found ( f g ) ( x ) ( f g ) ( x ) and now are asked to find ( f g ) ( −4 ) . ( f g ) ( −4 ) . ( f g ) ( x ) = x − 7 To find ( f g ) ( −4 ) , substitute x = −4 . ( f g ) ( −4 ) = −4 − 7 ( f g ) ( −4 ) = −11 ( f g ) ( x ) = x − 7 To find ( f g ) ( −4 ) , substitute x = −4 . ( f g ) ( −4 ) = −4 − 7 ( f g ) ( −4 ) = −11

Try It 5.87

For functions f ( x ) = x 2 − 5 x − 24 f ( x ) = x 2 − 5 x − 24 and g ( x ) = x + 3 , g ( x ) = x + 3 , find ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( −3 ) . ( f g ) ( −3 ) .

Try It 5.88

For functions f ( x ) = x 2 − 5 x − 36 f ( x ) = x 2 − 5 x − 36 and g ( x ) = x + 4 , g ( x ) = x + 4 , find ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( −5 ) . ( f g ) ( −5 ) .

Use the Remainder and Factor Theorem

Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as x − c , x − c , the value of the function at c , f ( c ) , c , f ( c ) , is the same as the remainder from the division problem.

Dividend	Divisor	Remainder	Function

		4		4
		3		3

To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend f ( x ) , f ( x ) , we multiply the quotient, q ( x ) q ( x ) times the divisor, x − c , x − c , and add the remainder, r .


If we evaluate this at we get:

This leads us to the Remainder Theorem.

Remainder Theorem

If the polynomial function f ( x ) f ( x ) is divided by x − c , x − c , then the remainder is f ( c ) . f ( c ) .

Example 5.45

Use the Remainder Theorem to find the remainder when f ( x ) = x 3 + 3 x + 19 f ( x ) = x 3 + 3 x + 19 is divided by x + 2 . x + 2 .

To use the Remainder Theorem, we must use the divisor in the x − c x − c form. We can write the divisor x + 2 x + 2 as x − ( −2 ) . x − ( −2 ) . So, our c c is −2 . −2 .

To find the remainder, we evaluate f ( c ) f ( c ) which is f ( −2 ) . f ( −2 ) .


To evaluate substitute
Simplify.

	The remainder is 5 when is divided by
Check: Use synthetic division to check.

The remainder is 5.

Try It 5.89

Use the Remainder Theorem to find the remainder when f ( x ) = x 3 + 4 x + 15 f ( x ) = x 3 + 4 x + 15 is divided by x + 2 . x + 2 .

Try It 5.90

Use the Remainder Theorem to find the remainder when f ( x ) = x 3 − 7 x + 12 f ( x ) = x 3 − 7 x + 12 is divided by x + 3 . x + 3 .

When we divided 8 a 3 + 27 8 a 3 + 27 by 2 a + 3 2 a + 3 in Example 5.41 the result was 4 a 2 − 6 a + 9 . 4 a 2 − 6 a + 9 . To check our work, we multiply 4 a 2 − 6 a + 9 4 a 2 − 6 a + 9 by 2 a + 3 2 a + 3 to get 8 a 3 + 27 8 a 3 + 27 .

Written this way, we can see that 4 a 2 − 6 a + 9 4 a 2 − 6 a + 9 and 2 a + 3 2 a + 3 are factors of 8 a 3 + 27 . 8 a 3 + 27 . When we did the division, the remainder was zero.

Whenever a divisor, x − c , x − c , divides a polynomial function, f ( x ) , f ( x ) , and resulting in a remainder of zero, we say x − c x − c is a factor of f ( x ) . f ( x ) .

The reverse is also true. If x − c x − c is a factor of f ( x ) f ( x ) then x − c x − c will divide the polynomial function resulting in a remainder of zero.

We will state this in the Factor Theorem.

Factor Theorem

For any polynomial function f ( x ) , f ( x ) ,

if x − c x − c is a factor of f ( x ) , f ( x ) , then f ( c ) = 0 f ( c ) = 0
if f ( c ) = 0 , f ( c ) = 0 , then x − c x − c is a factor of f ( x ) f ( x )

Example 5.46

Use the Remainder Theorem to determine if x − 4 x − 4 is a factor of f ( x ) = x 3 − 64 . f ( x ) = x 3 − 64 .

The Factor Theorem tells us that x − 4 x − 4 is a factor of f ( x ) = x 3 − 64 f ( x ) = x 3 − 64 if f ( 4 ) = 0 . f ( 4 ) = 0 . f ( x ) = x 3 − 64 To evaluate f ( 4 ) substitute x = 4 . f ( 4 ) = 4 3 − 64 Simplify. f ( 4 ) = 64 − 64 Subtract. f ( 4 ) = 0 f ( x ) = x 3 − 64 To evaluate f ( 4 ) substitute x = 4 . f ( 4 ) = 4 3 − 64 Simplify. f ( 4 ) = 64 − 64 Subtract. f ( 4 ) = 0

Since f ( 4 ) = 0 , f ( 4 ) = 0 , x − 4 x − 4 is a factor of f ( x ) = x 3 − 64 . f ( x ) = x 3 − 64 .

Try It 5.91

Use the Factor Theorem to determine if x − 5 x − 5 is a factor of f ( x ) = x 3 − 125 . f ( x ) = x 3 − 125 .

Try It 5.92

Use the Factor Theorem to determine if x − 6 x − 6 is a factor of f ( x ) = x 3 − 216 . f ( x ) = x 3 − 216 .

Access these online resources for additional instruction and practice with dividing polynomials.

Dividing a Polynomial by a Binomial
Synthetic Division & Remainder Theorem

Section 5.4 Exercises

Practice makes perfect.

Divide Monomials

In the following exercises, divide the monomials.

15 r 4 s 9 ÷ ( 15 r 4 s 9 ) 15 r 4 s 9 ÷ ( 15 r 4 s 9 )

20 m 8 n 4 ÷ ( 30 m 5 n 9 ) 20 m 8 n 4 ÷ ( 30 m 5 n 9 )

18 a 4 b 8 −27 a 9 b 5 18 a 4 b 8 −27 a 9 b 5

45 x 5 y 9 −60 x 8 y 6 45 x 5 y 9 −60 x 8 y 6

( 10 m 5 n 4 ) ( 5 m 3 n 6 ) 25 m 7 n 5 ( 10 m 5 n 4 ) ( 5 m 3 n 6 ) 25 m 7 n 5

( −18 p 4 q 7 ) ( −6 p 3 q 8 ) −36 p 12 q 10 ( −18 p 4 q 7 ) ( −6 p 3 q 8 ) −36 p 12 q 10

( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 2 b ) ( a 3 b ) ( 6 a 4 b 3 ) ( 4 a b 5 ) ( 12 a 2 b ) ( a 3 b )

( 4 u 2 v 5 ) ( 15 u 3 v ) ( 12 u 3 v ) ( u 4 v ) ( 4 u 2 v 5 ) ( 15 u 3 v ) ( 12 u 3 v ) ( u 4 v )

In the following exercises, divide each polynomial by the monomial.

( 9 n 4 + 6 n 3 ) ÷ 3 n ( 9 n 4 + 6 n 3 ) ÷ 3 n

( 8 x 3 + 6 x 2 ) ÷ 2 x ( 8 x 3 + 6 x 2 ) ÷ 2 x

( 63 m 4 − 42 m 3 ) ÷ ( −7 m 2 ) ( 63 m 4 − 42 m 3 ) ÷ ( −7 m 2 )

( 48 y 4 − 24 y 3 ) ÷ ( −8 y 2 ) ( 48 y 4 − 24 y 3 ) ÷ ( −8 y 2 )

66 x 3 y 2 − 110 x 2 y 3 − 44 x 4 y 3 11 x 2 y 2 66 x 3 y 2 − 110 x 2 y 3 − 44 x 4 y 3 11 x 2 y 2

72 r 5 s 2 + 132 r 4 s 3 − 96 r 3 s 5 12 r 2 s 2 72 r 5 s 2 + 132 r 4 s 3 − 96 r 3 s 5 12 r 2 s 2

10 x 2 + 5 x − 4 −5 x 10 x 2 + 5 x − 4 −5 x

20 y 2 + 12 y − 1 −4 y 20 y 2 + 12 y − 1 −4 y

Divide Polynomials using Long Division

In the following exercises, divide each polynomial by the binomial.

( y 2 + 7 y + 12 ) ÷ ( y + 3 ) ( y 2 + 7 y + 12 ) ÷ ( y + 3 )

( a 2 − 2 a − 35 ) ÷ ( a + 5 ) ( a 2 − 2 a − 35 ) ÷ ( a + 5 )

( 6 m 2 − 19 m − 20 ) ÷ ( m − 4 ) ( 6 m 2 − 19 m − 20 ) ÷ ( m − 4 )

( 4 x 2 − 17 x − 15 ) ÷ ( x − 5 ) ( 4 x 2 − 17 x − 15 ) ÷ ( x − 5 )

( q 2 + 2 q + 20 ) ÷ ( q + 6 ) ( q 2 + 2 q + 20 ) ÷ ( q + 6 )

( p 2 + 11 p + 16 ) ÷ ( p + 8 ) ( p 2 + 11 p + 16 ) ÷ ( p + 8 )

( 3 b 3 + b 2 + 4 ) ÷ ( b + 1 ) ( 3 b 3 + b 2 + 4 ) ÷ ( b + 1 )

( 2 n 3 − 10 n + 28 ) ÷ ( n + 3 ) ( 2 n 3 − 10 n + 28 ) ÷ ( n + 3 )

( z 3 + 1 ) ÷ ( z + 1 ) ( z 3 + 1 ) ÷ ( z + 1 )

( m 3 + 1000 ) ÷ ( m + 10 ) ( m 3 + 1000 ) ÷ ( m + 10 )

( 64 x 3 − 27 ) ÷ ( 4 x − 3 ) ( 64 x 3 − 27 ) ÷ ( 4 x − 3 )

( 125 y 3 − 64 ) ÷ ( 5 y − 4 ) ( 125 y 3 − 64 ) ÷ ( 5 y − 4 )

In the following exercises, use synthetic Division to find the quotient and remainder.

x 3 − 6 x 2 + 5 x + 14 x 3 − 6 x 2 + 5 x + 14 is divided by x + 1 x + 1

x 3 − 3 x 2 − 4 x + 12 x 3 − 3 x 2 − 4 x + 12 is divided by x + 2 x + 2

2 x 3 − 11 x 2 + 11 x + 12 2 x 3 − 11 x 2 + 11 x + 12 is divided by x − 3 x − 3

2 x 3 − 11 x 2 + 16 x − 12 2 x 3 − 11 x 2 + 16 x − 12 is divided by x − 4 x − 4

x 4 - 5 x 2 + 2 + 13 x + 3 x 4 - 5 x 2 + 2 + 13 x + 3 is divided by x + 3 x + 3

x 4 + x 2 + 6 x − 10 x 4 + x 2 + 6 x − 10 is divided by x + 2 x + 2

2 x 4 − 9 x 3 + 5 x 2 − 3 x − 6 2 x 4 − 9 x 3 + 5 x 2 − 3 x − 6 is divided by x − 4 x − 4

3 x 4 − 11 x 3 + 2 x 2 + 10 x + 6 3 x 4 − 11 x 3 + 2 x 2 + 10 x + 6 is divided by x − 3 x − 3

In the following exercises, divide.

For functions f ( x ) = x 2 − 13 x + 36 f ( x ) = x 2 − 13 x + 36 and g ( x ) = x − 4 , g ( x ) = x − 4 , find ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( −1 ) ( f g ) ( −1 )

For functions f ( x ) = x 2 − 15 x + 54 f ( x ) = x 2 − 15 x + 54 and g ( x ) = x − 9 , g ( x ) = x − 9 , find ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( −5 ) ( f g ) ( −5 )

For functions f ( x ) = x 3 + x 2 − 7 x + 2 f ( x ) = x 3 + x 2 − 7 x + 2 and g ( x ) = x − 2 , g ( x ) = x − 2 , find ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( 2 ) ( f g ) ( 2 )

For functions f ( x ) = x 3 + 2 x 2 − 19 x + 12 f ( x ) = x 3 + 2 x 2 − 19 x + 12 and g ( x ) = x − 3 , g ( x ) = x − 3 , find ⓐ ( f g ) ( x ) ( f g ) ( x ) ⓑ ( f g ) ( 0 ) ( f g ) ( 0 )

For functions f ( x ) = x 2 − 5 x + 2 f ( x ) = x 2 − 5 x + 2 and g ( x ) = x 2 − 3 x − 1 , g ( x ) = x 2 − 3 x − 1 , find ⓐ ( f · g ) ( x ) ( f · g ) ( x ) ⓑ ( f · g ) ( −1 ) ( f · g ) ( −1 )

For functions f ( x ) = x 2 + 4 x − 3 f ( x ) = x 2 + 4 x − 3 and g ( x ) = x 2 + 2 x + 4 , g ( x ) = x 2 + 2 x + 4 , find ⓐ ( f · g ) ( x ) ( f · g ) ( x ) ⓑ ( f · g ) ( 1 ) ( f · g ) ( 1 )

In the following exercises, use the Remainder Theorem to find the remainder.

f ( x ) = x 3 − 8 x + 7 f ( x ) = x 3 − 8 x + 7 is divided by x + 3 x + 3

f ( x ) = x 3 − 4 x − 9 f ( x ) = x 3 − 4 x − 9 is divided by x + 2 x + 2

f ( x ) = 2 x 3 − 6 x − 24 f ( x ) = 2 x 3 − 6 x − 24 divided by x − 3 x − 3

f ( x ) = 7 x 2 − 5 x − 8 f ( x ) = 7 x 2 − 5 x − 8 divided by x − 1 x − 1

In the following exercises, use the Factor Theorem to determine if x − c x − c is a factor of the polynomial function.

Determine whether x + 3 x + 3 a factor of x 3 + 8 x 2 + 21 x + 18 x 3 + 8 x 2 + 21 x + 18

Determine whether x + 4 x + 4 a factor of x 3 + x 2 − 14 x + 8 x 3 + x 2 − 14 x + 8

Determine whether x − 2 x − 2 a factor of x 3 − 7 x 2 + 7 x − 6 x 3 − 7 x 2 + 7 x − 6

Determine whether x − 3 x − 3 a factor of x 3 − 7 x 2 + 11 x + 3 x 3 − 7 x 2 + 11 x + 3

Writing Exercises

James divides 48 y + 6 48 y + 6 by 6 this way: 48 y + 6 6 = 48 y . 48 y + 6 6 = 48 y . What is wrong with his reasoning?

Divide 10 x 2 + x − 12 2 x 10 x 2 + x − 12 2 x and explain with words how you get each term of the quotient.

Explain when you can use synthetic division.

In your own words, write the steps for synthetic division for x 2 + 5 x + 6 x 2 + 5 x + 6 divided by x − 2 . x − 2 .

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/intermediate-algebra/pages/1-introduction

Authors: Lynn Marecek
Publisher/website: OpenStax
Book title: Intermediate Algebra
Publication date: Mar 14, 2017
Location: Houston, Texas
Book URL: https://openstax.org/books/intermediate-algebra/pages/1-introduction
Section URL: https://openstax.org/books/intermediate-algebra/pages/5-4-dividing-polynomials

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Divide Polynomial Worksheets

Related Topics: More Math Worksheets More Grade 7 Math Lessons Grade 7 Math Worksheets

There are three sets of polynomial worksheets:

Add & Subtract Polynomials
Multiply Polynomials
Divide Polynomials

Examples, solutions, videos, and worksheets to help Grade 7 and Grade 8 students learn how to divide polynomials.

How to divide polynomials?

There are six sets of dividing polynomials worksheets.

Divide Monomials
Divide Polynomials by Monomials
Divide Binomials by Binomials
Divide Trinomials by Binomials
Divide Polynomials by Binomials
Divide Polynomials (no remainder)

Here’s a step-by-step guide on how to divide polynomials using long division:

Write Down the Division: Write the division problem with the dividend (the polynomial being divided) inside the long division symbol and the divisor (the polynomial you’re dividing by) outside.
Divide the Leading Terms: Divide the leading term of the dividend by the leading term of the divisor. Write the result as the first term of the quotient.
Divide and Subtract: Divide the entire divisor by the first term of the quotient. Subtract the result from the dividend, and write the subtraction result below the line.
Bring Down: Bring down the next term from the dividend to the line.
Repeat: Repeat the process by dividing the new dividend term by the leading term of the divisor. Write the result as the next term of the quotient. Divide and subtract as before.
Continue: Continue this process until you have divided all the terms of the dividend. The final result is the quotient.
Remainder: If there’s a remainder after dividing all the terms, you can write it over the divisor as a fraction.

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Division of Polynomials

The following are some of the properties pertaining to fractions. These properties are discussed in Chapter 2.

1. a/b = ac/bc 2. (a+b)/c = a/c + b/c

3. a/b*c/d = ac/bd 4. a/b ÷ c/d = a/b * d/c

Note Since division by zero is not deﬁned, all denominators are assumed different from zero

First we will discuss division of monomials, then division of a polynomial by a monomial, and ﬁnally division of two polynomials.

Division of Monomials

From the properties of fractions and the rules governing exponents we have

a^8/a^5 = (a^5*a^3)/(a^5*1) = (a^3)/1 = a^3

= a^(8-5)

a^4/a^4 = 1

a^7/a^10 = (a^7*1)/(a^7*a^3) = 1/a^3

= 1/a^(10-7)

$Theorem 4 Division of Polynomial$

Proof a^m/a^n = (a^n * a^(m-n))/(a^n*1) = a^(m-n) When m>n

a^m/a^n = a^n/a^n = 1 When m=n

a^m/a^n = (a^m*1)/(a^m* a^(n-m)) = 1/a^(n-m) When m <strong>EXAMPLE </strong>1. [[2^6/2^2 = 2^(6-2) = 2^4

2. a^7/a^5 = a^(7-5) = a^2

3. (a - 1)^4/(a - 1)^3 = (a - 1)^(4-3) = (a - 1)

4. 5^4/5^4 = 1

5. (x +1)^3/(x + 1)^3 = 1

6. 3^8/3^12 = 1/3^(12-8) = 1/3^4

7. a^3/a^9 = 1/a^(9-3) = 1/a^6

8. (x + 2)^4/(x + 2)^6 = 1/((x + 2)^(6-4) = 1/(x + 2)^2

From the properties of fractions and the deﬁnition of exponents we have

(2/3)^4 = 2/3*2/3*2/3*2/3

= (2*2*2*2)/(3*3*3*3)

= 2^4/3^4

Let's see how our Polynomial solver simplifies this and similar problems. Click on "Solve Similar" button to see more examples.

$Theorem 5 Division of Polynomial$

Proof

$Proof for Theorem 5 Division of Polynomial$

then by the use of Theorem 2 and Theorem 3 and theorem 5, we have

((a^(m)b^(n))/(c^(p)d^(q)))^k = ((a^(m)b^(n))^k)/((c^(p)d^(q))^k = (a^(mk)b^(nk))/(c^(pk)d^(qk))

EXAMPLE Simplify (-30a^3b^2)/(12a^2b^4) by applying the laws of exponents.

Solution (-30a^3b^2)/(12a^2b^4) = -((2*3*5a^3b^2)/(2*2*3a^2b^4))

= -((2*3)/(2*2))*(5/2)*(a^3/a^2)*(b^2/b^4)

= -(5/2)*(a/1)*(1/b^2)

= -((5a)/(2b^2))

EXAMPLE By applying the laws of exponents , simplify [(2x^4yz)/(6xy^2)]^3

Solution We can simplify the fraction first before applying the outside exponent.

[(2x^4yz)/(6xy^2)]^3 = [(x^3z)/(3y)]^3 = (x^9z^3)/(3^3y^3) = (x^9z^3)/(27y^3)

EXAMPLE 12^4/18^3 = (2^2*3)^4/(2*3^2)^3 = (2^8*3^4)/(2^3*3^6)

= (2^8/2^3)*(3^4/3^6) = (2^5/1)*(1/(3^2)) = 32/9

EXAMPLE Simplify ((2a^2bc^3)^3)/((3ab^2)^2) by applying the laws of exponents

Solution Here we cannot simplify ﬁrst. since the numerator and the denominator have different powers. Apply the outside exponents ﬁrst. then simplify,

((2a^2bc^3)^3)/((3ab^2)^2) = (2^3a^6b^3c^9)/(3^2a^2b^4) = (8a^4c^9)/(9b)

EXAMPLE Perform the indicated operations and simplify:

16a^4b^3 ÷ (-2ab)^3 + 36a^5b^2 ÷ (-3a^2b)^2

Solution 16a^4b^3 ÷ (-2ab)^3 + 36a^5b^2 ÷ (-3a^2b)^2

= (16a^4b^3)/(-2ab)^3 + (36a^5b^2)/(-3a^2b)^2

= (16a^4b^3)/(-2^3a^3b^3) + (36a^5b^2)/(3^2a^4b^2)

= (16a^4b^3)/(-8a^3b^3) + (36a^5b^2)/(9a^4b^2)

= -2a + 4a = 2a

Division of a Polynomial by a Monomial

From the properties of fractions we have

$Division of Polynomial by a Monomial$

Keep in mind that

(a + b)/c means (a+b) ÷ c

$Condition for Division of Polynomial by a Monomial$

(a + b)/a = a/a + b/a = 1 + b/a

To divide a polynomial by a monomial, divide every term of the polynomial by the monomial.

EXAMPLE Divide (12x^3 - 6x^2 + 18x)/6x and simplify.

Solution (12x^3 - 6x^2 + 18x)/(6x) = (12x^3)/(6x) + (-6x^2)/(6x) + (18x)/(6x)

= 2x^2 - x + 3

EXAMPLE Divide (3a^3 - 2a^2b - ab^2)/(-ab) and simplify.

Solution (3a^3 - 2a^2b - ab^2)/(-ab) = (3a^3)/(-ab) + (-2a^2b)/(-ab) + (-ab^2)/(-ab)

= -((3a^2)/b) + 2a + b

(12a^4 + 4a^3 - 32a^2)/(4a^2) - (3a - 8)(a + 1)

Solution (12a^4 + 4a^3 - 32a^2)/(4a^2) - (3a - 8)(a + 1)

= (3a^2 + a - 8) - (3a^2 - 5a - 8)

= 3a^2 + a - 8 -3a^2 + 5a + 8 = 6a

Division of Two Polynomials

Division is deﬁned as the inverse operation of multiplication; thus we start with a multiplication problem and then deduce the division operation.

(x^2 + 3x -5)(2x -7) = x^2(2x - 7) + 3x(2x - 7) + (-5)(2x - 7)

= 2x^3 - 7x^2) + (6x^2 -21x) + (-10x + 35)

= 2x^3 - x^2 - 31x + 35

Hence if (2x^3 - x^2 - 31x + 35) is divided by (2x - 7) , the result is (x^2 + 3x - 5) , the first polynomial of multiplication problem.

The polynomial (2x^3 - x^2 - 31x + 35) is called the dividend , (2x -7) is called the divisor , and (x^2 + 3x -5) is called the quotient . The first term of the dividend, 2x^3 , comes from multiplying the first term of quotient, x^2 , by the first term of the divisor, 2x . Thus to obtain the first term of quotient, x^2 , we divide the first term of the dividend , 2x^3 , by the first term of the divisor, 2x . Multiplying the entire divisor (2x -7) by that ﬁrst term of the quotient, x^2 , we get 2x^3 - 7x^2 . Subtracting 2x^3 - 7x^2 from the dividend,

(2x^3 - x^2 - 31x + 35) - (2x^3 - 7x^2) = 6x^2 - 31x + 35

The quantity 6x^2 - 31x + 35 is our new dividend. The ﬁrst term, 6x^2 , of the new dividend comes from multiplying the second term of the quotient, 3x , by the first term of the divisor, 2x . Thus to get the second term of the quotient, 3x , divide the ﬁrst term of the new dividend, 6x^2 , by the ﬁrst term of the divisor, 2x . Multiplying the divisor (2x - 7) by the second term of the quotient, 3x , we get 6x^2 - 21x . Subtracting 6x^2 - 21x from the new dividend,

(6x^2 -31x +35) - (6x^2 - 21x) = -10x + 35

The quantity -10x + 35 is the new dividend. Dividing the first term of the new dividend (-10x) by the first term of the divisor, 2x , we get the third term of the quotient (-5) . Multiplying the divisor (2x-7) by the third term of the quotient (-5) , we get -10x + 35 . Subtracting (-10x + 35) from the dividend (-10x + 35) , We get zero.

Let us start the problem again an arrange it in a manner analogous to that of long division in arithmetic.

$Rearrange problem for division of Polynomials$

Hence (2x^3 - x^2 - 31x +35)/(2x - 7) = x^2 + 3x -5

DEFINITION The degree of a polynomial in a literal number is the greatest exponent of that literal number in the polynomial.

EXAMPLE x^5y - 7x^4y^2 - 2x^3y^3 + 9y^4 is a polynomial of degree 5 in x and degree 4 in y .

To divide two polynomials, we start by arranging the terms of the dividend according to the decreasing exponents of one of the literals, leaving spaces for the missing powers (including terms with zero coefﬁcients for the missing terms). Arrange the terms of the divisor also according to the decreasing exponents of the same literal used in arranging the terms of the dividend. Divide the ﬁrst term of the dividend by the ﬁrst term of the divisor to get the ﬁrst term of the quotient. Multiply the ﬁrst term of the quotient by each term of the divisor and write the product under the like terms in the dividend. Subtract the product from the dividend to arrive at a new dividend.

To ﬁnd the next term and all subsequent terms of the quotient, treat the new dividend as if it were the original dividend. Continue this procedure until you get zero or until the degree of the newly derived polynomial. with respect to the literal used in arranging the dividend, is at least one degree less than the degree of the divisor in that literal.

The last polynomial is called the remainder .

EXAMPLE Divide (6x^3 - 17x^2 + 16) by (3x - 4) .

Solution Write the dividend as 6x^3 - 17x^2 + 0x + 16

$Remainder from Polynomial Division$

EXAMPLE Divide (19x^2 - 10x^3 + x^5 - 14x + 6) by (x^2 + 1 -2x) .

Solution Write the dividend as x^5 + 0x^4 - 10x^3 + 19x^2 - 14x + 6 .

Write the divisor as x^2 -2x + 1

$Getting the remainder$

Hence (x^5 - 10x^3 + 19x^2 -14x + 6)/(x^2 - 2x + 1)

= x^3 + 2x^2 - 7x + 3 + (-x + 3)/(x^2 - 2x + 1)

= x^3 + 2x^2 - 7x + 3 - (x -3)/(x^2 - 2x + 1)

Note This form is similar to the form used in arithmetic when we write

20/7 = 2 + 6/7

EXAMPLE Divide (2x^4 - 3y^4 - 13x^2y^2 + 14xy^3) by (x^2 + 2xy - 3y^2)

Solution Write the dividend as 2x^4 + 0x^3y - 13x^2y^2 + 14xy^3 - 3y^4 .

$How to simplify problem for Division of polynomials$

Hence (2x^4 - 13x^2y^2 + 14xy^3 - 3y^4)/(x^2 + 2xy -3y^2) = 2x^2 - 4xy + y^2

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Dividing Polynomials

Dividing polynomials is an arithmetic operation where we divide a polynomial by another polynomial, generally with a lesser degree as compared to the dividend. The division of two polynomials may or may not result in a polynomial. Let's learn about dividing polynomials in this article in detail.

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What is Dividing Polynomials?

Polynomials are algebraic expressions that consist of variables and coefficients . It is written in the following format: 5x 2 + 6x - 17. This polynomial has three terms that are arranged according to their degree. The term with the highest degree is placed first, followed by the lower ones. Dividing polynomials is an algorithm to solve a rational number that represents a polynomial divided by a monomial or another polynomial. The divisor and the dividend are placed exactly the same way as we do for regular division. For example, if we need to divide 5x 2 + 7x + 25 by 6x - 25, we write it in this way:

\[\dfrac{(5 x ^2+7 x+25)}{(6 x -25)}\]

The polynomial written on top of the bar is the numerator ( 5x 2 + 7x + 25), while the polynomial written below the bar is the denominator (6x - 25). This can be understood by the following figure which shows that the numerator becomes the dividend and the denominator becomes the divisor.

Dividing Polynomials by Monomials

While dividing polynomials by monomials , the division can be done in two ways. One is by simply separating the '+' and '-' operator signs. That means, we break the polynomial from the operating sign, and solve each part separately. Another method is to do the simple factorization and further simplifying. Let us have a look at both methods in detail.

Splitting the Terms Method

Split the terms of the polynomial separated by the operator ( '+' or '-' ) between them and simplify each term. For example, (4x 2 - 6x) ÷ (2x) can be solved as shown here. We first take common terms from the numerators and denominators of both the terms, we get, [(4x 2 ) / (2x)] - [(6x) / (2x)]. Canceling the common term 2x from the numerator and the denominator, we get 2x - 3.

Factorization Method

When you divide polynomials you may have to factor the polynomial to find a common factor between the numerator and the denominator. For example: Divide the following polynomial: (2x 2 + 4x) ÷ 2x. Both the numerator and denominator have a common factor of 2x. Thus, the expression can be written as 2x(x + 2) / 2x. Canceling out the common term 2x, we get x+2 as the answer.

Dividing Polynomials by Binomials

For dividing polynomials by binomials or any other type of polynomials , the most common and general method is the long division method . When there are no common factors between the numerator and the denominator, or if you can't find the factors, you can use the long division process to simplify the expression.

Dividing Polynomials Using Long Division

Let us go through the algorithm of dividing polynomials by binomials using an example: Divide: (4x 2 - 5x - 21) ÷ (x - 3). Here, (4x 2 - 5x - 21) is the dividend, and (x - 3) is the divisor which is a binomial. Observe the division shown below, followed by the steps.

Step 1. Divide the first term of the dividend (4x 2 ) by the first term of the divisor (x), and put that as the first term in the quotient (4x).

Step 2. Multiply the divisor by that answer, place the product (4x 2 - 12x) below the dividend.

Step 3. Subtract to create a new polynomial (7x - 21).

Step 4. Repeat the same process with the new polynomial obtained after subtraction.

So, when we are dividing a polynomial (4x 2 - 5x - 21) with a binomial (x - 3), the quotient is 4x+7 and the remainder is 0.

Dividing Polynomials Using Synthetic Division

Synthetic division is a technique to divide a polynomial with a linear binomial by only considering the values of the coefficients. In this method, we first write the polynomials in the standard form from the highest degree term to the lowest degree terms. While writing in descending powers, use 0's as the coefficients of the missing terms. For example, x 3 +3 has to be written as x 3 + 0x 2 + 0x + 3. Follow the steps given below for dividing polynomials using the synthetic division method:

Let us divide x 2 + 3 by x - 4.

Step 1: Write the divisor in the form of x - k and write k on the left side of the division. Here, the divisor is x-4, so the value of k is 4.

Step 2: Set up the division by writing the coefficients of the dividend on the right and k on the left. [Note: Use 0's for the missing terms in the dividend]

Step 3: Now, bring down the coefficient of the highest degree term of the dividend as it is. Here, the leading coefficient is 1 (coefficient of x 2 ).

Step 4: Multiply k with that leading coefficient and write the product below the second coefficient from the left side of the dividend. So, we get, 4×1=4 that we will write below 0.

Step 5: Add the numbers written in the second column. Here, by adding we get 0+4=4.

Step 6: Repeat the same process of multiplication of k with the number obtained in step 5 and write the product in the next column to the right.

Step 7: At last, we will write the final answer which will be one degree less than the dividend. So, here, in our dividend, the highest degree term is x 2 , therefore, in the quotient, the highest degree term will be x. Therefore, the answer obtained is x+4+(19/x-4).

dividing polynomials using synthetic division

Topics Related to Dividing Polynomials

Check these articles to know more about the concept of dividing polynomials and its related topics.

Long division of polynomials
Synthetic division of polynomial
Division Algorithm for Polynomials
Dividing Two Polynomials
Division of Polynomial by Linear Factor

Dividing Polynomials Examples

Example 1: Alex is stuck on a problem while working on dividing polynomials. Can you help him to obtain the quotient: (x 4 - 10x 3 + 27x 2 - 46x + 28) ÷ (x - 7).

Therefore, the quotient is x 3 - 3x 2 + 6x - 4.

Example 2: Stacy needs help in finding the remainder while dividing polynomials. Can you help her solve this?

(4x 3 + 5x 2 + 5x + 8) ÷ (4x + 1)

Solution: Using the long division method of dividing polynomials, we get,

Therefore, the remainder is 7.

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Practice Questions on Dividing Polynomials

Faqs on dividing polynomials.

Dividing polynomials is one of the arithmetic operations performed on two given polynomials. In this, the dividend is generally of a higher degree and the divisor is a lower degree polynomial .

Why is Dividing Polynomials Important?

Dividing polynomials is important because it provides an algorithm to solve a rational number that represents a polynomial divided by a monomial or another polynomial.

What is the Easiest Way for Dividing Polynomials?

The easiest way to divide polynomials is by using the long division method. However, in the case of the division of polynomials by a monomial, it can be directly solved by splitting the terms or by factorization.

What are the Two Methods of Dividing Polynomials?

The two methods to divide polynomials are given below:

Synthetic division method
Long division method

What is Polynomial Division Used for in Real-Life?

We use polynomial division for various aspects of our day-to-day lives. We need it for coding, engineering, designing, architecting, and various other real-life areas.

What Method of Dividing Polynomials by a Monomial is Best?

In the case of the division of polynomials by a monomial, it can be directly solved by splitting the terms or by factorization. We can divide each term of the dividend with the given monomial and find the result.

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When you see a polynomial that is a fraction composed of two polynomials — one as the numerator and the other as the denominator — it can often be simplified using long division. The procedure is based on the same principle of long division for whole numbers. Here is how it works. The word "polynomial" means "many terms" — something like is a common polynomial form. Note that while polynomials can contain constants (such as or ), variables (such as and ) and exponents, such as or , polynomials must not contain negative exponents or division by a variable such as . Let’s try some polynomial division practice. Consider this polynomial: First, we rewrite this as a form of long division. The only difference from regular long divisions is that, instead of numbers, they are polynomials. Divide by , which gives . Write this on the line above . Multiply by , which gives . Write this below . Subtract from to give . Write this below and carry the as well. Divide by to give . Write this next to above the line. Multiply by to give . Write this below . Subtract from to give . Write this below and carry the again. Divide by to give . Write this next to above the line. Multiply by to give . Write this below . Subtract from to give a remainder of . Write the final answer: Interested in learning more about polynomial division? Start with our polynomial division problems at the top of this page. Our practice questions let you tackle problems at your own pace. If you get stumped, click on "Solution" to see exactly how we arrived at the answer. Want even more help? Sign up for today. At Cymath, it is our mission to help students tackle math concepts via . Try our online explanations and solutions to boost your math competence today!

Polynomials - Long Division

A polynomial looks like this:

Polynomials can sometimes be divided using the simple methods shown on Dividing Polynomials .

But sometimes it is better to use "Long Division" (a method similar to Long Division for Numbers )

Numerator and Denominator

We can give each polynomial a name:

the top polynomial is the numerator
the bottom polynomial is the denominator

If you have trouble remembering, think denominator is down- ominator.

Write it down neatly:

the denominator goes first,
then a ")",
then the numerator with a line above

Both polynomials should have the "higher order" terms first (those with the largest exponents , like the "2" in x 2 ).

Divide the first term of the numerator by the first term of the denominator, and put that in the answer.
Multiply the denominator by that answer, put that below the numerator
Subtract to create a new polynomial

It is easier to show with an example!

Write it down neatly like below, then solve it step-by-step (press play):

Check the answer:

Multiply the answer by the bottom polynomial, we should get the top polynomial:

The previous example worked perfectly, but that is not always so! Try this one:

After dividing we were left with "2", this is the "remainder".

The remainder is what is left over after dividing.

But we still have an answer: put the remainder divided by the bottom polynomial as part of the answer, like this:

"Missing" Terms

There can be "missing terms" (example: there may be an x 3 , but no x 2 ). In that case either leave gaps, or include the missing terms with a coefficient of zero.

Write it down with "0" coefficients for the missing terms, then solve it normally (press play):

See how we needed a space for "3x 3 " ?

More than One Variable

So far we have been dividing polynomials with only one variable ( x ), but we can handle polynomials with two or more variables (such as x and y ) using the same method.

Math Article
Polynomial Division

Polynomial Division & Long Division Algorithm

The polynomial division involves the division of one polynomial by another. The division of polynomials can be between two monomials, a polynomial and a monomial or between two polynomials. Before learning how to divide polynomials, let’s have a brief introduction to the definition of polynomial and its related terms .

Polynomial:

A polynomial is an algebraic expression of the type a n x n + a n−1 x n−1 +…………………a 2 x 2 + a 1 x + a 0 , where “n” is either 0 or positive variables and real coefficients.

In this expression, a n , a n−1 …..a 1 , a 0 are coefficients of the terms of the polynomial.

The highest power of x in the above expression, i.e. n is known as the degree of the polynomial.

If p(x) represents a polynomial and x = k such that p(k) = 0 then k is the root of the given polynomial.

Table of Contents:

Division of Polynomials

Types of Polynomial Division

Monomial by Monomial
Polynomial by Monomial
Polynomial by Binomial
Polynomial by another Polynomial
Division algorithm
Practice problems

Given a polynomial equation, p(x)=x –x–2. Find the zeroes of the equation.

Given Polynomial, p(x)=x –x–2

Zeroes of the equation is given by:

x –2x+x–2=0

x(x−2)+1(x–2)

(x+1)(x−2)=0

⇒ x=−1

Or, x=2

Thus, -1 and 2 are zeroes of the given polynomial.

It is to be noted that the highest power(degree) of the polynomial gives the maximum number of zeroes of the polynomial.

Division of Polynomial

The division is the process of splitting a quantity into equal amounts. In terms of mathematics, the process of repeated subtraction or the reverse operation of multiplication is termed as division. For example, when 20 is divided by 4 we get 5 as the result since 4 is subtracted 5 times from 20.

The four basic operations viz. addition, subtraction, multiplication and division can also be performed on algebraic expressions . Let us understand the process and different methods of dividing polynomials and algebraic expressions .

For dividing polynomials, generally, three cases can arise:

Division of a monomial by another monomial

Division of a polynomial by monomial.

Division of a polynomial by binomial
Division of a polynomial by another polynomial

Let us discuss all these cases one by one:

Consider the algebraic expression 40x 2 is to be divided by 10x then

40x 2 /10x = (2×2×5×2×x×x)/(2×5×x)

Here, 2, 5 and x are common in both the numerator and the denominator.

Hence, 40x 2 /10x = 4x

The second case is when a polynomial is to be divided by a monomial. For dividing polynomials, each term of the polynomial is separately divided by the monomial (as described above) and the quotient of each division is added to get the result. Consider the following example:

– 12xy + 9x by 3x.
The given expression 24x – 12xy + 9x has three terms viz. 24x , – 12xy and 9x. For dividing the polynomial with a monomial, each term is separately divided as shown below:(24x –12xy+9x)/3x = (24x /3x)–(12xy/3x)+(9x/3x) = 8x –4y+3

Division of a Polynomial by Binomial

As we know, binomial is an expression with two terms. Dividing a polynomial by binomial can be done easily. Here, first we need to write the given polynomial in standard form. Now, using the long division method, we can divide the polynomial as given below.

– 8x + 5 by x – 1.

The Dividend is 3x – 8x + 5 and the divisor is x – 1.

After this, the leading term of the dividend is divided by the leading term of the divisor i.e. 3x ÷ x =3x .

This result is multiplied by the divisor i.e. 3x (x -1) = 3x -3x and it is subtracted from the divisor.

Now again, this result is treated as a dividend and the same steps are repeated until the remainder becomes “0” or its degree becomes less than that of the divisor as shown below.

Division of Polynomial by Another Polynomial

For dividing a polynomial with another polynomial, the polynomial is written in standard form i.e. the terms of the dividend and the divisor are arranged in decreasing order of their degrees. The method to solve these types of divisions is “Long division”. In algebra, an algorithm for dividing a polynomial by another polynomial of the same or lower degree is called polynomial long division. It is the generalised version of the familiar arithmetic technique called long division. Let us take an example.

Example: Divide x 2 + 2x + 3x 3 + 5 by 1 + 2x + x 2 .

Let us arrange the polynomial to be divided in the standard form.

3x 3 + x 2 + 2x + 5

Divisor = x 2 + 2x + 1

Using the method of long division of polynomials, let us divide 3x 3 + x 2 + 2x + 5 by x 2 + 2x + 1.

Step 1: To obtain the first term of the quotient, divide the highest degree term of the dividend, i.e. 3x 3 by the highest degree term of the divisor, i.e. x 2 .

3x 3 /x 2 = 3x

Now, carry out the division process.

Step 2: Now, to obtain the second term of the quotient, divide the highest degree term of the new dividend, i.e. –5x 2 by the highest degree term of the divisor, i.e. x 2 .

-5x 2 /x 2 = -5

Again carry out the division process with – 5x 2 – x + 5 (the remainder in the previous step).

Step 3: The remainder obtained from the previous step is 9x + 10.

The degree of 9x + 10 is less than the divisor x 2 + 2x + 1. So, we cannot continue the division any further.

Dividing polynomial by another polynomial

Polynomial Division Algorithm

If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x)

r(x) = 0 or degree of r(x) < degree of g(x)

This result is called the Division Algorithm for polynomials.

From the previous example, we can verify the polynomial division algorithm as:

p(x) = 3x 3 + x 2 + 2x + 5

g(x) = x 2 + 2x + 1

Also, quotient = q(x) = 3x – 1

remainder = r(x) = 9x + 10

g(x) × q(x) + r(x) = (x 2 + 2x + 1) × (3x – 5) + (9x + 10)

= 3x 3 + 6x 2 + 3x – 5x 2 – 10x – 5 + 9x + 10

= 3x 3 + x 2 + 2x + 5

Hence, the division algorithm is verified.

Polynomial Division Questions

If the polynomial x 4 – 6x 3 + 16x 2 – 25x + 10 is divided by another polynomial x 2 – 2x + k, the remainder comes out to be x + a, find k and a.
Divide the polynomial 2t 4 + 3t 3 – 2t 2 – 9t – 12 by t 2 – 3.
Find all the zeroes of 2x 4 – 3x 3 – 3x 2 + 6x – 2, if two of its zeroes are √2 and −√2.

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6-week online workshop, 6-week proportional reasoning course, virtual summit, support / help, how to teach algebra: division of polynomials.

Factoring and dividing polynomials (in senior level courses) are usually skills we tell students: “here are the steps to factor” or “here’s the steps to use long division to divide these two polynomials”.

Why are we teaching these as separate skills when in fact they are laying in a progression of division stretching back to middle school and elementary school.

Let’s use prior understanding and connect it to new understanding so students see one skill instead of something new to memorize.

In this video we’ll show you how to connect factoring and polynomial division to existing models for division from elementary and middle school.

When you use these visual modelling techniques with your students you’ll see greater improvement on retention and understanding of factoring and division.

In particular, you’ll learn:

Why showing your students the progression of division of dividing multi-digit numbers through division of polynomials in algebra will improve their understanding and competence in algebra.
How to connect different representations of division and factoring to fuel sense making in your students.
How and why you should be using the area model to factor polynomials and/or divide polynomials.

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x^{\msquare}

\log_{\msquare}

\sqrt{\square}

\nthroot[\msquare]{\square}

\le

\ge

\frac{\msquare}{\msquare}

\cdot

\div

x^{\circ}

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\square\frac{\square}{\square}

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x^{\msquare}

\log_{\msquare}

\sqrt{\square}

\nthroot[\msquare]{\square}

\le

\ge

\frac{\msquare}{\msquare}

\cdot

\div

x^{\circ}

\pi

\left(\square\right)^{'}

\frac{d}{dx}

\frac{\partial}{\partial x}

\int

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\lim

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- \twostack{▭}{▭}	\lt	7	8	9	\div	AC
+ \twostack{▭}{▭}	\gt	4	5	6	\times	\square\frac{\square}{\square}
\times \twostack{▭}{▭}	\left(	1	2	3	-	x
▭\:\longdivision{▭}	\right)	.	0	=	+	y

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long\:division\:\frac{x^{4}+6x^{2}+2}{x^{2}+5}
long\:division\:\frac{x^{3}+x^{2}}{x^{2}+x-2}
long\:division\:\frac{4x^{3}-7x^{2}-11x+5}{4x+5}
long\:division\:\frac{2x^{2}+5x-18}{(x+4)}
How do you divide polynomials with long division?
To divide polynomials using long division, divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by the quotient term, subtract the result from the dividend, bring down the next term of the dividend, and repeat the process until there is a remainder of lower degree than the divisor. Write the quotient as the sum of all the quotient terms and the remainder as the last polynomial obtained.
What is the formula for polynomial division?
Given two polynomials f(x) and g(x), where the degree of g(x) is less than or equal to the degree of f(x), the polynomial division of f(x) by g(x) can be expressed by the formula: f(x)/g(x) = q(x) + r(x)/g(x), where q(x) is the quotient polynomial, and r(x) is the remainder polynomial.
What are the 2 methods to divide polynomials?
The two common methods for dividing polynomials: long division and synthetic division.

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High School Math Solutions – Polynomials Calculator, Dividing Polynomials In the last post, we talked about how to multiply polynomials. In this post, we will talk about to divide polynomials....

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1. Multiple Choice Edit 1 minute 1 pt (x 2 -3x+2)/(x-2) x+1 x-2 x-1 2x+2
2. Multiple Choice Edit 1 minute 1 pt (x 3 +2x 2 -2x-1)/(x-1) x 2 +3x-1 x 2 +5x+1 x 2 +3x+1 x 2 -3x+1
3. Multiple Choice Edit 1 minute 1 pt (3x 2 +3x-18)/(x-2) (3x+9) (3x-9) (9x-3) (9x+3)
4. Multiple Choice Edit 1 minute 1 pt Divide (x 2 + 5x - 8) by (x + 3). What is the remainder? 16 12 -14 -16
6. Multiple Choice Edit 1 minute 1 pt (2x 3 + 5x 2 + 9) ÷ (x + 3) 2x 2 - x + 3 2x 4 - x 2 + 3x 2x 2 + 11x + 33 2x 2 +x +3
7. Multiple Choice Edit 1 minute 1 pt (4x 2 - 24x + 35) ÷ (2x - 5) 2x - 7 2x - 12 2x + 12 2x + 7
8. Multiple Choice Edit 1 minute 1 pt Divide using synthetic division (2x 3 + 5x 2 + 9) ÷ (x + 3) 2x 2 - x + 3 2x 3 - x 2 + 3x 2x 2 + 11x + 33 + 108/x+3 2x 2 - 5x + 12
9. Multiple Choice Edit 1 minute 1 pt Divide using synthetic division: (x 3 - 11x 2 +19x + 67) ÷ (x - 7) x 2 - 4x - 9 + 4/(x-7) x 2 - 4x - 10 + 2/(x-7) x 2 - 4x - 12 + 5/(x-7) x 2 - 4x - 12 + 1/(x-7)
10. Multiple Choice Edit 1 minute 1 pt (3x 3 + 5x - 1) ÷ (x + 1) 3x 3 - 3x 2 + 8x - 9 3x 2 - 3x + 8 - 9/(x+1) 3x 2 + 3x + 8 + 7/(x+1) 3x 3 + 3x 2 + 8x + 7
11. Multiple Choice Edit 1 minute 1 pt 2x 3 - 5x 2 + 3x + 7 ÷ x - 2 2x 3 - x 2 + x + 9 2x 2 - x + 1 2x 2 - x + 1 + 9/x-2 2x 2 - 9x - 15 - 23/x-2
12. Multiple Choice Edit 1 minute 1 pt x 3 -3x 2 -10x+24 ÷ x+3 x 2 -7x+5 x-8 x 2 -6x+8 x+9
13. Multiple Choice Edit 1 minute 12 pts Give the correct name for 2x 3 +5x Linear Binomial Cubic Binomial Quadratic Trinomial Linear Trinomial
14. Multiple Choice Edit 1 minute 1 pt Divide: (9x 2 + 6) / (3x) 3x + 6/3x 3x + 2 3 + 6/3x 3x + 6
15. Multiple Choice Edit 1 minute 1 pt The area of a rectangular pool table is: 4x 4 + 24x 2 +40x. The length is 4x. What is the width? x 4 + 6x 2 + 10x 20x 2 + 36x x 3 + 6x + 10

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Division Facts and Word Problems

Join us on a mathematical journey as we explore division facts and problem-solving strategies. This lesson is designed to help students strengthen their understanding of division, practice division facts, and apply their knowledge to solve word problems. Through engaging activities and real-world scenarios, students will develop confidence in their division skills and enhance their problem-solving abilities. Get ready to dive into the world of division!

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COMMENTS

Division of Polynomials
Step 3: We multiply or distribute the answer obtained in the previous step by the polynomial in front of the division symbol. Step 4: We subtract the obtained expression and write the next term. Step 5: We repeat steps 2, 3, and 4 until there are no more terms remaining. Step 6: We write the final answer. The remaining term after the last terms ...
3.5e: Exercises
1) If division of a polynomial by a binomial results in a remainder of zero, what can be conclude? 2) If a polynomial of degree $n$ is divided by a binomial of degree $1$, what is the degree of the quotient? Answers to odd exercises: 1. The binomial is a factor of the polynomial.
Polynomial Long Division
Remember that example 1 is a division of polynomial with three terms (trinomial) by a binomial. Hopefully, you see a slight difference. : Focus on the leftmost terms of both the dividend and divisor. : Divide the leftmost term of the dividend by the leftmost term of the divisor. : Place their product under the dividend.
3.5: Dividing Polynomials
The Division Algorithm tells us that a polynomial dividend can be written as the product of the divisor and the quotient added to the remainder. Synthetic division is a shortcut that can be used to divide a polynomial by a binomial in the form $x−k.$ Polynomial division can be used to solve application problems, including area and volume.
Polynomial Division Questions
As we know, the polynomial division is one of the important concepts of Class 10 maths. Polynomial division questions and answers are given here to help students learn the division of polynomials by a monomial, binomial and another polynomial. In this article, you will get solved questions on polynomial division and some practice questions.
5.4 Dividing Polynomials
Given a polynomial and a binomial, use long division to divide the polynomial by the binomial. Set up the division problem. Determine the first term of the quotient by dividing the leading term of the dividend by the leading term of the divisor. Multiply the answer by the divisor and write it below the like terms of the dividend.
Dividing Polynomials
Dividing. Sometimes it is easy to divide a polynomial by splitting it at the "+" and "−" signs in the top part, like this (press play): When the polynomial was split into parts we still had to keep the "/3" under each one. Then the highlighted parts were "reduced" (6/3 = 2 and 3/3 = 1) to leave the answer of 2x-1.
5.4 Dividing Polynomials
The same problem in the synthetic division format is shown next. Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the x x and x 2 x 2 are removed. as well as the − x 2 − x 2 and −4 x −4 x as they are opposite the term above.
Polynomial division
After we have added, subtracted, and multiplied polynomials, it's time to divide them! This will prove to be a little bit more sophisticated. It turns out that not every polynomial division results in a polynomial. When it doesn't, we end up with a remainder (just like with integer division!).
Divide Polynomial Worksheets
There are six sets of dividing polynomials worksheets. Here's a step-by-step guide on how to divide polynomials using long division: Write Down the Division: Write the division problem with the dividend (the polynomial being divided) inside the long division symbol and the divisor (the polynomial you're dividing by) outside.
Division of Polynomials Step-by-Step Math Problem Solver
a+ba = aa+ba = 1+ba. To divide a polynomial by a monomial, divide every term of the polynomial by the monomial. EXAMPLE Divide 12 x3-6 x2+18 x6 x and simplify. Solution 12 x3-6 x2+18 x6 x = 12 x36 x+− 6 x26 x+18 x6 x. = 2 x2-x+3. Let's see how our Polynomial solver simplifies this and similar problems.
6.6: Divide Polynomials
Divide a Polynomial by a Monomial. In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.. The method we'll use to divide a polynomial by a monomial is based on the properties of fraction addition.
Dividing Polynomials
For example, x 3 +3 has to be written as x 3 + 0x 2 + 0x + 3. Follow the steps given below for dividing polynomials using the synthetic division method: Let us divide x 2 + 3 by x - 4. Step 1: Write the divisor in the form of x - k and write k on the left side of the division. Here, the divisor is x-4, so the value of k is 4.
Cool math Algebra Help Lessons: Division of Polynomials
Dividing by Monomials. Polynomials: Substituting for X Cruncher. Long Division. Long Division - Freaky Things That Can Happen Part 1. Long Division - Freaky Things That Can Happen Part 2.
Algebra Polynomial division
Consider this polynomial: First, we rewrite this as a form of long division. The only difference from regular long divisions is that, instead of numbers, they are polynomials. Step 1: Divide. Write this on the line above. Step 2: Multiply. Write this below. Step 3: Subtract. Write this below.
Polynomials
Then: Divide the first term of the numerator by the first term of the denominator, and put that in the answer. Multiply the denominator by that answer, put that below the numerator. Subtract to create a new polynomial. Repeat, using the new polynomial. It is easier to show with an example! Example: x2 − 3x − 10 x + 2.
Polynomial Division & Long Division Algorithm
Polynomial Division is the division of a polynomial by a monomial, binomial or another polynomial using different methods. ... Division algorithm; Practice problems; Example: Given a polynomial equation, p(x)=x 2 -x-2. Find the zeroes of the equation. ... The method to solve these types of divisions is "Long division". In algebra, an ...
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How To Teach Algebra: Division of Polynomials
How To Teach Algebra: Division of Polynomials. Factoring and dividing polynomials (in senior level courses) are usually skills we tell students: "here are the steps to factor" or "here's the steps to use long division to divide these two polynomials". Why are we teaching these as separate skills when in fact they are laying in a ...
Polynomial Long Division Calculator
Given two polynomials f (x) and g (x), where the degree of g (x) is less than or equal to the degree of f (x), the polynomial division of f (x) by g (x) can be expressed by the formula: f (x)/g (x) = q (x) + r (x)/g (x), where q (x) is the quotient polynomial, and r (x) is the remainder polynomial. The two common methods for dividing ...
Dividing Polynomials
Dividing Polynomials quiz for 11th grade students. Find other quizzes for Mathematics and more on Quizizz for free! ... Is this division problem worked correctly? This is correct! This is incorrect! 6. Multiple Choice. Edit. 1 minute. 1 pt (2x 3 + 5x 2 + 9) ÷ (x + 3) 2x 2 - x + 3. 2x 4 - x 2 + 3x. 2x 2 + 11x + 33.
Solve Polynomial Equations: Practice Problems & Solutions
Ace your courses with our free study and lecture notes, summaries, exam prep, and other resources
Solving Math Problems with GEMDAS & Polynomial Functions
Mathematics document from University of Santo Tomas, 11 pages, Grade 10- Mathematics Week 1: GEMDAS & Polynomial Functions Name: Grade and Section: Objectives Review how to solve problems using GEMDAS. Review polynomial functions and the degrees. Perform division of polynomials using long division and synthet
Division Facts and Word Problems
Join us on a mathematical journey as we explore division facts and problem-solving strategies. This lesson is designed to help students strengthen their understanding of division, practice division facts, and apply their knowledge to solve word problems. Through engaging activities and real-world scenarios, students will develop confidence in ...
HyperCase: Join a multi-disciplinary team to solve problems in
HyperCase is a healthcare startup incubator bringing together law, business, engineering and medical students to develop innovative solutions in healthcare. HyperCase aims to teach entrepreneurship through a product development curriculum, board meetings, guest speakers and an end-of-year conference ...