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Math (SAT®)Math (SAT®)35 views·Updated May 27, 2026·3 pages

Mastering Synthetic Division of Polynomials

user profile picture
Natalie Fritz@atalieritz_kziv

Dividing polynomials might seem challenging at first, but it's actually... Show more

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# 3.2.A: Long Division of Polynomials Notes - Divide a polynomial $f(x)$ by a divisior $d(x)$ to get a quotient $q(x)$ and remainder $r(x)$

Long Division of Polynomials

Just like dividing regular numbers, polynomial long division helps you split one polynomial by another. The goal is to find a quotient and remainder that satisfy: f(x)d(x)=q(x)+r(x)d(x)\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}. Always remember that the remainder's degree must be less than the divisor's degree.

When setting up polynomial long division, organize it like regular long division but leave placeholders (zeros) for any missing terms. For example, when dividing $16x^3+8x^2+29x+7by by 4x+1,youstartbydividingthefirstterms(, you start by dividing the first terms (16x^3 ÷ 4x = 4x^2$) and continue the process systematically.

Through the division process, you multiply, subtract, bring down terms, and repeat until the remainder's degree is less than the divisor's. In our example, the final answer is $4x^2+x+7+\frac{9}{4x+1}$, showing both the quotient and remainder parts.

Try This! When dividing polynomials, make sure to line up terms with the same powers of x. This organization makes the subtraction steps much easier and helps prevent errors.

2
of 3
# 3.2.A: Long Division of Polynomials Notes - Divide a polynomial $f(x)$ by a divisior $d(x)$ to get a quotient $q(x)$ and remainder $r(x)$

Synthetic Division of Polynomials

Synthetic division is a shortcut method that works when dividing any polynomial by a linear factor in the form (x−k)(x-k). This technique uses a compact arrangement of just the coefficients, making the process much faster than long division.

To set up synthetic division, place the value of kk from the divisor $x-k$ on the left of your work. Write all the coefficients of the dividend in a row (including zeros for missing terms). Then follow the pattern of bringing down the first coefficient, multiplying by kk, adding down the column, and repeating.

The Remainder Theorem tells us that if a polynomial f(x)f(x) is divided by x−kx-k, then f(k)f(k) equals the remainder. Related to this, the Factor Theorem states that x−kx-k is a factor of polynomial f(x)f(x) if and only if f(k)=0f(k) = 0. This gives us a quick way to check if something is a factor!

Important Connection: When the remainder equals zero in synthetic division, the divisor (x−k)(x-k) is definitely a factor of the original polynomial. This also means that kk is a solution to the polynomial equation.

3
of 3
# 3.2.A: Long Division of Polynomials Notes - Divide a polynomial $f(x)$ by a divisior $d(x)$ to get a quotient $q(x)$ and remainder $r(x)$

More Synthetic Division Examples

Synthetic division saves time, especially with more complex polynomials. For example, when dividing $2x^3-15x^2+21x+24by by x-3,wesetupthesyntheticdivisionwith, we set up the synthetic division with k=3andworkthroughtheprocess.Theresultshowsaquotientof and work through the process. The result shows a quotient of 2x^2-9x-6witharemainderof with a remainder of 6,writtenas, written as 2x^2-9x-6+\frac{6}{x-3}$.

When using synthetic division, always remember to include zeros for any missing terms in your polynomial. For instance, in $2x^4-7x^2-4x+2dividedby divided by x+1,weneedtoincludeazeroforthemissing, we need to include a zero for the missing x^3$ term before starting the division process.

The final answer format depends on whether the divisor is a factor. If the remainder is zero as in the fifth example where $3x^4-25x^3+27x^2+6x+7$ divided by $x-7$ gives a remainder of 0, then the divisor is a factor and the answer is just the quotient. Otherwise, write the answer with the remainder over the divisor.

Pro Tip: When dividing by (x+a)(x+a) instead of (x−k)(x-k), use k=−ak=-a in your synthetic division. For example, to divide by (x+1)(x+1), use k=−1k=-1 in your setup.

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Math (SAT®)Math (SAT®)35 views·Updated May 27, 2026·3 pages

Mastering Synthetic Division of Polynomials

user profile picture
Natalie Fritz@atalieritz_kziv

Dividing polynomials might seem challenging at first, but it's actually a systematic process that follows clear steps. In these notes, we'll explore two methods for dividing polynomials: long division and synthetic division, which give you powerful tools to solve polynomial... Show more

1
of 3
# 3.2.A: Long Division of Polynomials Notes - Divide a polynomial $f(x)$ by a divisior $d(x)$ to get a quotient $q(x)$ and remainder $r(x)$

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Long Division of Polynomials

Just like dividing regular numbers, polynomial long division helps you split one polynomial by another. The goal is to find a quotient and remainder that satisfy: f(x)d(x)=q(x)+r(x)d(x)\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}. Always remember that the remainder's degree must be less than the divisor's degree.

When setting up polynomial long division, organize it like regular long division but leave placeholders (zeros) for any missing terms. For example, when dividing $16x^3+8x^2+29x+7by by 4x+1,youstartbydividingthefirstterms(, you start by dividing the first terms (16x^3 ÷ 4x = 4x^2$) and continue the process systematically.

Through the division process, you multiply, subtract, bring down terms, and repeat until the remainder's degree is less than the divisor's. In our example, the final answer is $4x^2+x+7+\frac{9}{4x+1}$, showing both the quotient and remainder parts.

Try This! When dividing polynomials, make sure to line up terms with the same powers of x. This organization makes the subtraction steps much easier and helps prevent errors.

2
of 3
# 3.2.A: Long Division of Polynomials Notes - Divide a polynomial $f(x)$ by a divisior $d(x)$ to get a quotient $q(x)$ and remainder $r(x)$

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

Synthetic Division of Polynomials

Synthetic division is a shortcut method that works when dividing any polynomial by a linear factor in the form (x−k)(x-k). This technique uses a compact arrangement of just the coefficients, making the process much faster than long division.

To set up synthetic division, place the value of kk from the divisor $x-k$ on the left of your work. Write all the coefficients of the dividend in a row (including zeros for missing terms). Then follow the pattern of bringing down the first coefficient, multiplying by kk, adding down the column, and repeating.

The Remainder Theorem tells us that if a polynomial f(x)f(x) is divided by x−kx-k, then f(k)f(k) equals the remainder. Related to this, the Factor Theorem states that x−kx-k is a factor of polynomial f(x)f(x) if and only if f(k)=0f(k) = 0. This gives us a quick way to check if something is a factor!

Important Connection: When the remainder equals zero in synthetic division, the divisor (x−k)(x-k) is definitely a factor of the original polynomial. This also means that kk is a solution to the polynomial equation.

3
of 3
# 3.2.A: Long Division of Polynomials Notes - Divide a polynomial $f(x)$ by a divisior $d(x)$ to get a quotient $q(x)$ and remainder $r(x)$

Sign up to see the content. It's free!

  • Access to all documents
  • Improve your grades
  • Join milions of students

More Synthetic Division Examples

Synthetic division saves time, especially with more complex polynomials. For example, when dividing $2x^3-15x^2+21x+24by by x-3,wesetupthesyntheticdivisionwith, we set up the synthetic division with k=3andworkthroughtheprocess.Theresultshowsaquotientof and work through the process. The result shows a quotient of 2x^2-9x-6witharemainderof with a remainder of 6,writtenas, written as 2x^2-9x-6+\frac{6}{x-3}$.

When using synthetic division, always remember to include zeros for any missing terms in your polynomial. For instance, in $2x^4-7x^2-4x+2dividedby divided by x+1,weneedtoincludeazeroforthemissing, we need to include a zero for the missing x^3$ term before starting the division process.

The final answer format depends on whether the divisor is a factor. If the remainder is zero as in the fifth example where $3x^4-25x^3+27x^2+6x+7$ divided by $x-7$ gives a remainder of 0, then the divisor is a factor and the answer is just the quotient. Otherwise, write the answer with the remainder over the divisor.

Pro Tip: When dividing by (x+a)(x+a) instead of (x−k)(x-k), use k=−ak=-a in your synthetic division. For example, to divide by (x+1)(x+1), use k=−1k=-1 in your setup.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Similar Content

More

Most popular content: Synthetic Division

1

Most popular content in Math (SAT®)

9

Most popular content

9

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.6/5App Store
4.7/5Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan SiOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha KlichAndroid user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

AnnaiOS user