\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Precalculus - Synthetic Division

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Synthetic Division

Synthetic division is a short-cut technique to long division of polynomials. Synthetic division cannot be used in all cases but, when it can be used, it may be quicker than long division.

When To Use Synthetic Division

Use synthetic division when the denominator is of the form \( x + c \) or \( ax - b \), i.e. we are dividing by a first order polynomial (the highest power of the variable is one).
DO NOT USE synthetic division when the denominator is anything other than a first order polynomial. For example, if the denominator contains something like \( x^2 + 3x + 1 \), synthetic division will not work.

How To Do Synthetic Division

Here is a great video explaining, step-by-step, how to do synthetic division using a specific example. She goes through the example slowly with lots of explanation.

Synthetic Division... How?

Synthetic Division Remainder

Before you even go through the synthetic division steps, you can find out what the remainder will be. Let's say we have \( x-a \) in the denominator. Now normally we don't want zero in the denominator, right? That is a big no-no! However, in this case, we are going to set this equal to zero and solve for \(x\), i.e. \( x - a = 0 \to x = a\). Now we plug this x-value into the numerator and the result is the remainder after synthetic division. Pretty cool, eh? This is a great way to check your answer.

Synthetic Division Compared To Long Division

Here is a great video comparing long division to synthetic division side-by-side using an example. We recommend that you watch this especially to compare how synthetic division parallels long division. We believe you will understand both techniques better after watching this video.

Learn Math Tutorials - Synthetic Division vs. Long Division [6min-26secs]

Alternative Technique

Most instructors teach synthetic division like we've discussed so far. However, there is an alternative technique that we recommend. We will show the video explaining this technique, first with no remainder and then explain how to handle a remainder.
Note - Check with your instructor to make sure they allow you to use this technique.

Alternative Technique With No Remainder

First, we will look at the case when the divisor is a factor of the dividend (the denominator is a factor of the numerator), i.e. there is no remainder. This will allow you to understand the technique without the added complication of a remainder.

This video shows a cool way to do synthetic division and really learn it (despite the name of the video). Take a minute and watch it.

Dr James Tanton - Synthetic Division: How to understand It by not doing it. [10min-45secs]

video by Dr James Tanton

Okay, so you watched the video, right? Let's expand on it a bit before discussing what we do when we have a remainder. There are some things you need to watch out for.

You need a column and row for every factor.
For example, if you have the a numerator polynomial of \(2x^3 +2x-6\), you need 4 columns, even one for the \(x^2\) with coefficient zero. The same goes for a denominator polynomial. You would need a row for the factor with a zero coefficient.

You do not need the x's
He mentions this in the video, but if you want to just use the coefficients, you can. You will just need to have some kind of system to keep track of terms. We will use the x's all the time on this site. Leaving them off just introduces something else you have to keep track of and you have enough to deal with already. So carrying them along is not that big of a deal.

Alternative Technique With Remainder

Okay, so you watched the video and you probably think that this technique works only when there is no remainder. What do you do when the remainder is not zero? Before you read on, try to figure out how to do this on your own. It is intuitive and not difficult.

The way we came up with to handle remainders is to build the boxes with an extra column. When you do this and try an example, you will figure out that the boxes in that extra column contain the remainder and that the box in the lower right corner will never be used. So you can put a dash in that lower right corner box, gray it out or just leave it blank (although if you leave it blank, it might not be clear that you have finished the problem). Is this the same idea as you came up with? If not, let us know if you found a better way.

When you have a remainder, you need to watch for the same things as you did when you didn't have a remainder that we discussed above. In fact, since you don't know if your answer will have a remainder, it will help you to always have an extra column of boxes. Then, if the column is all zero, you know you do not have a remainder. When we do this, we label that column R to make sure it is clear that the factors in those boxes are the remainder.

Expanded Synthetic Division

There are some videos on YouTube showing how to do synthetic division when your denominator has higher powers. We do not recommend that you learn those techniques unless your instructor wants you to. You won't need them most of the time and they are almost as complicated as long division. So we recommend long division for all cases that cannot be covered by basic synthetic division discussed on this page. This is, of course, only our opinion, so use your own judgement.

Okay, time for the practice problems.

Practice

Unless otherwise instructed, divide the polynomials as indicated, giving your answers in exact form.

\(\displaystyle{ \frac{2x^3-5x+14}{x+3} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{2x^3-5x+14}{x+3} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 2x^2 - 6x + 13 - \frac{25}{x+3} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{2x^3-5x+14}{x+3} }\) using synthetic division, giving your answer in exact form.

Solution

In this video, he performs long division first and then uses synthetic division.

2790 video

Final Answer

\(\displaystyle{ 2x^2 - 6x + 13 - \frac{25}{x+3} }\)

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\(\displaystyle{ \frac{y^3-6}{y+2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{y^3-6}{y+2} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ y^2-2y+4 - \frac{14}{y+2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{y^3-6}{y+2} }\) using synthetic division, giving your answer in exact form.

Solution

In this video, he performs long division first and then uses synthetic division.

2791 video

Final Answer

\(\displaystyle{ y^2-2y+4 - \frac{14}{y+2} }\)

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\(\displaystyle{ \frac{x^3-2x^2-5x+6}{x-3} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3-2x^2-5x+6}{x-3} }\) using synthetic division, giving your answer in exact form.

Final Answer

\( x^2 + x - 2 \)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3-2x^2-5x+6}{x-3} }\) using synthetic division, giving your answer in exact form.

Solution

2792 video

Final Answer

\( x^2 + x - 2 \)

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\(\displaystyle{ \frac{x^3+5x^2+7x+2}{x+2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+5x^2+7x+2}{x+2} }\) using synthetic division, giving your answer in exact form.

Final Answer

\( x^2 + 3x + 1 \)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+5x^2+7x+2}{x+2} }\) using synthetic division, giving your answer in exact form.

Solution

2793 video

Final Answer

\( x^2 + 3x + 1 \)

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\(\displaystyle{ \frac{x^3+3x^2-4x-12}{x+3} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+3x^2-4x-12}{x+3} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ x^2 - 4 }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+3x^2-4x-12}{x+3} }\) using synthetic division, giving your answer in exact form.

Solution

2794 video

Final Answer

\(\displaystyle{ x^2 - 4 }\)

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\(\displaystyle{ \frac{x^3+x^2-2x-8}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+x^2-2x-8}{x-2} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ x^2+3x+4 }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+x^2-2x-8}{x-2} }\) using synthetic division, giving your answer in exact form.

Solution

2795 video

Final Answer

\(\displaystyle{ x^2+3x+4 }\)

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\(\displaystyle{ \frac{3x^3 - 5x^2 + x -2}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^3 - 5x^2 + x -2}{x-2} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 3x^2 + x + 3 + \frac{4}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^3 - 5x^2 + x -2}{x-2} }\) using synthetic division, giving your answer in exact form.

Solution

2796 video

Final Answer

\(\displaystyle{ 3x^2 + x + 3 + \frac{4}{x-2} }\)

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\(\displaystyle{ \frac{3x^2+7x-20}{x+5} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^2+7x-20}{x+5} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 3x - 8 + \frac{20}{x+5} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^2+7x-20}{x+5} }\) using synthetic division, giving your answer in exact form.

Solution

2797 video

Final Answer

\(\displaystyle{ 3x - 8 + \frac{20}{x+5} }\)

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\(\displaystyle{ \frac{7x^3+6x-8}{x-4} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{7x^3+6x-8}{x-4} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 7x^2 + 28x + 118 + \frac{464}{x-4} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{7x^3+6x-8}{x-4} }\) using synthetic division, giving your answer in exact form.

Solution

2798 video

Final Answer

\(\displaystyle{ 7x^2 + 28x + 118 + \frac{464}{x-4} }\)

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\(\displaystyle{ \frac{3x^4-5x^2+6}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^4-5x^2+6}{x-2} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 3x^3 + 6x^2 + 7x + 14 + \frac{34}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^4-5x^2+6}{x-2} }\) using synthetic division, giving your answer in exact form.

Solution

2799 video

Final Answer

\(\displaystyle{ 3x^3 + 6x^2 + 7x + 14 + \frac{34}{x-2} }\)

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\(\displaystyle{ \frac{3x^4-2x^3+5x^2+8}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^4-2x^3+5x^2+8}{x-2} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 3x^3 + 4x^2 + 13x + 26 + \frac{60}{x-2} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{3x^4-2x^3+5x^2+8}{x-2} }\) using synthetic division, giving your answer in exact form.

Solution

2800 video

Final Answer

\(\displaystyle{ 3x^3 + 4x^2 + 13x + 26 + \frac{60}{x-2} }\)

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\(\displaystyle{ \frac{x^3+5x^2-3x+4}{x-3} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+5x^2-3x+4}{x-3} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ x^2 + 8x + 21 + \frac{67}{x-3} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{x^3+5x^2-3x+4}{x-3} }\) using synthetic division, giving your answer in exact form.

Solution

2801 video

Final Answer

\(\displaystyle{ x^2 + 8x + 21 + \frac{67}{x-3} }\)

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\(\displaystyle{ \frac{2x^3-3x^2+5x-8}{x-4} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{2x^3-3x^2+5x-8}{x-4} }\) using synthetic division, giving your answer in exact form.

Final Answer

\(\displaystyle{ 2x^2 +5x +25 + \frac{92}{x-4} }\)

Problem Statement

Evaluate \(\displaystyle{ \frac{2x^3-3x^2+5x-8}{x-4} }\) using synthetic division, giving your answer in exact form.

Solution

At the beginning of this video, he calculates the remainder and then compares it with the result of synthetic division. This is a great way to check your answer.

2802 video

Final Answer

\(\displaystyle{ 2x^2 +5x +25 + \frac{92}{x-4} }\)

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\(\displaystyle{\frac{x^3-2x^2+3x-4}{x-2}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{x^3-2x^2+3x-4}{x-2}}\) using synthetic division, giving your answer in exact form.

Solution

1501 video

video by PatrickJMT

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\(\displaystyle{\frac{x^4-x^2+5}{x+3}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{x^4-x^2+5}{x+3}}\) using synthetic division, giving your answer in exact form.

Solution

1502 video

video by PatrickJMT

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\(\displaystyle{\frac{x^3+8x^2-17x+15}{x+2}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{x^3+8x^2-17x+15}{x+2}}\) using synthetic division, giving your answer in exact form.

Solution

1505 video

video by MIP4U

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\(\displaystyle{\frac{x^3-3x^2+4x-7}{x-3}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{x^3-3x^2+4x-7}{x-3}}\) using synthetic division, giving your answer in exact form.

Solution

1506 video

video by MIP4U

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\(\displaystyle{\frac{2x^3+6x^2+29}{x+4}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{2x^3+6x^2+29}{x+4}}\) using synthetic division, giving your answer in exact form.

Solution

1507 video

video by MIP4U

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\(\displaystyle{\frac{2x^3+6x^2-17x+15}{x+5}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{2x^3+6x^2-17x+15}{x+5}}\) using synthetic division, giving your answer in exact form.

Solution

1508 video

video by MIP4U

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\(\displaystyle{\frac{y^5-32}{y-2}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{y^5-32}{y-2}}\) using synthetic division, giving your answer in exact form.

Solution

1509 video

video by MIP4U

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\(\displaystyle{\frac{3x^2-5x+1}{x+4}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{3x^2-5x+1}{x+4}}\) using synthetic division, giving your answer in exact form.

Solution

1514 video

video by Your Math Gal

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\(\displaystyle{\frac{2x^2-13x+10}{x-3}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{2x^2-13x+10}{x-3}}\) using synthetic division, giving your answer in exact form.

Solution

1515 video

video by Your Math Gal

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\(\displaystyle{\frac{2x^3-x^2+4x+1}{x-3}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{2x^3-x^2+4x+1}{x-3}}\) using synthetic division, giving your answer in exact form.

Solution

1516 video

video by Your Math Gal

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\(\displaystyle{\frac{3x-2x^3+5}{x+2}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{3x-2x^3+5}{x+2}}\) using synthetic division, giving your answer in exact form.

Solution

1517 video

video by Your Math Gal

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\(\displaystyle{\frac{3x^2+2x-3}{x+4}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{3x^2+2x-3}{x+4}}\) using synthetic division, giving your answer in exact form.

Solution

1520 video

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\(\displaystyle{\frac{x^3-5x^2-x+5}{x-1}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{x^3-5x^2-x+5}{x-1}}\) using synthetic division, giving your answer in exact form.

Solution

1521 video

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\(\displaystyle{\frac{3x^3+4x^2-2x-1}{x+4}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{3x^3+4x^2-2x-1}{x+4}}\) using synthetic division, giving your answer in exact form.

Solution

1503 video

video by Khan Academy

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\(\displaystyle{\frac{2x^5-x^3+3x^2-2x+7}{x-3}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{2x^5-x^3+3x^2-2x+7}{x-3}}\) using synthetic division, giving your answer in exact form.

Solution

1504 video

video by Khan Academy

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\(\displaystyle{\frac{16x^3-x+14x-12x^2}{2x+1}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{16x^3-x+14x-12x^2}{2x+1}}\) using synthetic division, giving your answer in exact form.

Solution

1510 video

video by MIP4U

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\(\displaystyle{\frac{2x^2-9x+8}{2x-3}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{2x^2-9x+8}{2x-3}}\) using synthetic division, giving your answer in exact form.

Solution

1511 video

video by Your Math Gal

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\(\displaystyle{\frac{5-23x+12x^2}{4x-1}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{5-23x+12x^2}{4x-1}}\) using synthetic division, giving your answer in exact form.

Solution

1512 video

video by Your Math Gal

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\(\displaystyle{\frac{7-3x^2+9x^3}{3x+2}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{7-3x^2+9x^3}{3x+2}}\) using synthetic division, giving your answer in exact form.

Solution

1513 video

video by Your Math Gal

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\(\displaystyle{\frac{3x^6-11x^5-7x^4+18x^3-15x^2-37x+4}{x-4}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{3x^6-11x^5-7x^4+18x^3-15x^2-37x+4}{x-4}}\) using synthetic division, giving your answer in exact form.

Solution

1519 video

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\(\displaystyle{\frac{9x^6+15x^5-6x^4+x^2+x-2}{x+2}}\)

Problem Statement

Evaluate \(\displaystyle{\frac{9x^6+15x^5-6x^4+x^2+x-2}{x+2}}\) using synthetic division, giving your answer in exact form.

Solution

1518 video

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