Get started

Math (SAT®)43Updated Sep 7, 20263 pages

Mastering Synthetic Division of Polynomials

Report
user profile picture
Natalie Fritz@atalieritz_kziv
Add to Google sources
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 problems efficiently.
Synthetic Division of Polynomials – page 1

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

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 16x3+8x2+29x+716x^3+8x^2+29x+7 by 4x+14x+1, you start by dividing the first terms (16x3÷4x=4x216x^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 4x2+x+7+94x+14x^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.

Synthetic Division of Polynomials – page 2

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

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−kx-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.

Synthetic Division of Polynomials – page 3

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

More Synthetic Division Examples

Synthetic division saves time, especially with more complex polynomials. For example, when dividing 2x3−15x2+21x+242x^3-15x^2+21x+24 by x−3x-3, we set up the synthetic division with k=3k=3 and work through the process. The result shows a quotient of 2x2−9x−62x^2-9x-6 with a remainder of 66, written as 2x2−9x−6+6x−32x^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 2x4−7x2−4x+22x^4-7x^2-4x+2 divided by x+1x+1, we need to include a zero for the missing x3x^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 3x4−25x3+27x2+6x+73x^4-25x^3+27x^2+6x+7 divided by x−7x-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...

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.

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

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

Similar Content

Algebra 2

Synthetic Division

Using synthetic division to dived polynomials.

11th18611
Algebra 2

Algebra 2 Ch. 5 Polynomials

Dividing polynomials, synthetic division, evaluating fractions, cubic factoring, fundamental theorem

11th1211
Algebra 1

Synthetic Division

Notes on how to accurately do synthetic division

9th672
Algebra 2

Synthetic Divison

Will help you on the notes step by step toutorial

23450
Algebra 2

Set and Interval Notation

Learn how to complete a table, graph solution sets, and state domain and range using interval notation in inequality notation.

10th1122
Geometry

Triangle Congruencee by SSS and SAS

Notes about triangle congruence using theorems SSS and SAS

10th540109
See more

Most popular content: Math (SAT®)

Math (SAT®)

Calculating percentages

Percentage and Percent Change - Problem Solving and Data Analysis - SAT Math

9th2051
Math (SAT®)

EVERY FORMULA YOU NEED FOR SAT

Learn every formula for DSAT MATH!

10th58524

Most popular content

Algebra

FAR NOTES- CPM

Compiled by CPM

College2,13847
Fun Stuff

DMV Practice Test 1

First set of Questions from DMV Handbook

1,413131
AP Physics 1

Newton's Second Law of Motion

A detailed explanation of Concept of Newton's Second Law of Motion, Examples, Formulas, and Sample Problems with solution.

44414
Algebra

FAR NOTES - KUYA WOWOWIE

Far notes by Kuya Wowowie

College97218
Algebra

FAR NOTES-HERCULES

LAST MINUTE NOTES BY HERCULES CPA

College1,21018
Algebra

RFBT NOTES- HERCULES

LAST MINUTE NOTES BY HERCULES CPA

College92616
Physics

Translational Motion and Rotational Motion

Applications of Translational and Rotational Motion

11th1821
Algebra

AFAR NOTES- CPM

Compiled by CPM

College1,32717
English

Introduction to linguistics

Introduction to linguistics exam revision notes. Structure of language, typologies of language, parts of speech, language families, Chomsky, Hockett, semantic triangle, Prague Linguistic Circle, writing systems, acquisition and learning

1,30253

Students love us, and so will you.

4.6/5App Store
4.7/5Google Play
Stefan SiOS user

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.

Samantha KlichAndroid 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.

AnnaiOS 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.