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The Remainder and Factor Theorems

Author: Sophia
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what's covered
In this lesson, you will learn two very important theorems that are useful in analyzing polynomial functions. Specifically, this lesson will cover:

Table of Contents

1. Evaluating a Polynomial Using the Remainder Theorem

In the last section, we saw that dividing a polynomial f open parentheses x close parentheses by d open parentheses x close parentheses produced the following result:

f open parentheses x close parentheses equals d open parentheses x close parentheses q open parentheses x close parentheses plus r open parentheses x close parentheses comma

where d open parentheses x close parentheses is the divisor, q open parentheses x close parentheses is the quotient, and r open parentheses x close parentheses is the remainder, and the degree of r open parentheses x close parentheses is less than the degree of d open parentheses x close parentheses. When the divisor is x minus k comma a linear expression, the remainder must be a constant.

Then, we can write f open parentheses x close parentheses equals open parentheses x minus k close parentheses q open parentheses x close parentheses plus r.

This being said, let’s use this expression to evaluate f open parentheses k close parentheses colon

table attributes columnalign left end attributes row cell f open parentheses k close parentheses equals open parentheses k minus k close parentheses q open parentheses k close parentheses plus r end cell row cell space space space space space space equals 0 times q open parentheses k close parentheses plus r end cell row cell space space space space space space equals r end cell end table

This result is summarized by the remainder theorem: when polynomial f open parentheses x close parentheses is divided by x minus k comma the remainder is f open parentheses k close parentheses.

This means we have two ways to find the remainder when f open parentheses x close parentheses is divided by x minus k colon

  1. Evaluate f open parentheses k close parentheses.
  2. Perform long division or synthetic division and note the remainder.

EXAMPLE

Consider the function f open parentheses x close parentheses equals x cubed minus 4 x squared plus 7 x minus 6. Use the remainder theorem to evaluate f open parentheses 3 close parentheses.

For this example, we’ll use synthetic division.

table attributes columnalign left end attributes row cell table row 3 row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 cell short dash 4 end cell 7 cell short dash 6 end cell row blank 3 cell short dash 3 end cell 12 end table close with bar below end cell row cell space space space space space space 1 space space short dash 1 space space space space space 4 space space space space space 6 end cell end table

Thus, the remainder is 6.

To check this:

table attributes columnalign left end attributes row cell f open parentheses 3 close parentheses equals 3 cubed minus 4 open parentheses 3 close parentheses squared plus 7 open parentheses 3 close parentheses minus 6 end cell row cell space space space space space space equals 27 minus 36 plus 21 minus 6 end cell row cell space space space space space space equals 6 end cell end table

try it
Consider the function f open parentheses x close parentheses equals 4 x to the power of 4 minus 3 x squared plus 11 x minus 20.
Use the remainder theorem to evaluate f   (5).
To use the remainder theorem, use synthetic division. Here is what the table should look like:

table row 5 row blank end table stack open vertical bar table attributes columnalign right end attributes row 4 0 cell short dash 3 end cell 11 cell short dash 20 end cell row blank cell space 20 end cell 100 485 2480 end table close with bar below space space space space space space 4 space space space 20 space space space space 97 space space space 496 space space 2460

The remainder is 2460, which means that f open parentheses 5 close parentheses equals 2460.

The remainder theorem might seem like a roundabout way to evaluate a function, but the connection between f open parentheses k close parentheses and the remainder is what is important. This is explored in the next section of this tutorial.

term to know
Remainder Theorem
When polynomial f open parentheses x close parentheses is divided by x minus k comma the remainder is f open parentheses k close parentheses.

2. Using the Factor Theorem to Solve a Polynomial Equation

When f open parentheses x close parentheses is divided by x minus k and the remainder is 0, we also know f open parentheses k close parentheses equals 0. This means that k is a zero of f open parentheses x close parentheses. This also means that if f open parentheses k close parentheses equals 0 comma then open parentheses x minus k close parentheses is a factor of f open parentheses x close parentheses.

We should define the factor theorem: the number k is a zero of f open parentheses x close parentheses if and only if open parentheses x minus k close parentheses is a factor of f open parentheses x close parentheses.

So, why use division to find the remainder if it is just equal to f open parentheses k close parentheses ? When using division to find the remainder, we not only get the remainder, we also get the quotient when f open parentheses x close parentheses is divided by x minus k. When the remainder is equal to 0, this is very useful since we then have two factors of f open parentheses x close parentheses colon the quotient, and x minus k.

EXAMPLE

Show that x plus 2 is a factor of f open parentheses x close parentheses equals x cubed minus 6 x squared minus x plus 30. Then, write f open parentheses x close parentheses in completely factored form, if possible.

First, use synthetic division with k equals short dash 2.

table attributes columnalign left end attributes row cell table row cell short dash 2 end cell row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 cell short dash 6 end cell cell short dash 1 end cell 30 row blank cell short dash 2 end cell 16 cell short dash 30 end cell end table close with bar below end cell row cell space space space space space space space space space 1 space space short dash 8 space space space 15 space space space space space space space 0 end cell end table

The remainder is 0, which means that x plus 2 is a factor of f open parentheses x close parentheses.

This means f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses open parentheses x squared minus 8 x plus 15 close parentheses. Notice the quadratic factor can be factored even further.

In completely factored form, f open parentheses x close parentheses equals open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses open parentheses x minus 5 close parentheses.

try it
Consider the function f open parentheses x close parentheses equals x cubed plus 4 x squared minus 4 x minus 16.
Use the remainder theorem to show that (x - 2) is a factor of f   (x  ).
Use synthetic division with x equals 2. The table is below.

table row 2 row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 4 cell short dash 4 end cell cell short dash 16 end cell row blank 2 12 16 end table close with bar below space space space space space space 1 space space 6 space space space space space 8 space space space space space space space 0

Since the remainder is 0, x equals 2 is a zero of f open parentheses x close parentheses.
Write f   (x  ) in completely factored form.
Since x equals 2 is a zero of f open parentheses x close parentheses comma then open parentheses x minus 2 close parentheses is a factor. The last row of the synthetic division tells us that the quotient is x squared plus 6 x plus 8. Putting this together, we have f open parentheses x close parentheses equals open parentheses x minus 2 close parentheses open parentheses x squared plus 6 x plus 8 close parentheses.

Next, in factored form, x squared plus 6 x plus 8 equals open parentheses x plus 4 close parentheses open parentheses x plus 2 close parentheses.

This means that the completely factored form of f open parentheses x close parentheses is f open parentheses x close parentheses equals open parentheses x minus 2 close parentheses open parentheses x plus 2 close parentheses open parentheses x plus 4 close parentheses.

Taking things one step further, the factor theorem can be used to find the zeros of a polynomial function.

watch
In this video, we'll apply the factor and remainder theorems and walk through finding the solutions of a cubic equation.

EXAMPLE

Find all zeros of f open parentheses x close parentheses equals 3 x cubed plus x squared minus 20 x plus 12, given that x plus 3 is a factor of f open parentheses x close parentheses.

Since we know x plus 3 is a factor of f open parentheses x close parentheses comma use synthetic division with k equals short dash 3 colon

table attributes columnalign left end attributes row cell table row cell short dash 3 end cell row blank end table stack open vertical bar table attributes columnalign right end attributes row 3 1 cell short dash 20 end cell 12 row blank cell short dash 9 end cell 24 cell short dash 12 end cell end table close with bar below end cell row cell space space space space space space space space space 3 space space short dash 8 space space space space space space space 4 space space space space space space space 0 end cell end table

Now, write f open parentheses x close parentheses in factored form: f open parentheses x close parentheses equals open parentheses x plus 3 close parentheses open parentheses 3 x squared minus 8 x plus 4 close parentheses

Now, solve open parentheses x plus 3 close parentheses open parentheses 3 x squared minus 8 x plus 4 close parentheses equals 0.

open parentheses x plus 3 close parentheses open parentheses 3 x squared minus 8 x plus 4 close parentheses equals 0 This is the original equation.
open parentheses x plus 3 close parentheses open parentheses 3 x minus 2 close parentheses open parentheses x minus 2 close parentheses equals 0 Factor the quadratic factor.
x plus 3 equals 0 comma space 3 x minus 2 equals 0 comma space x minus 2 equals 0 Set each factor equal to 0.
x equals short dash 3 comma space x equals 2 over 3 comma space x equals 2 Solve each equation for x.

Thus, the zeros of f open parentheses x close parentheses are -3, 2 over 3 comma and 2.

Note: to solve 3 x squared minus 8 x plus 4 equals 0 comma another method such as the quadratic formula could have been used if factoring wasn’t apparent. This is highlighted in the next example.

EXAMPLE

Find all zeros of f open parentheses x close parentheses equals x cubed plus 3 x squared minus 11 x plus 2 comma given that x minus 2 is a factor of f open parentheses x close parentheses.

Since we know x minus 2 is a factor of f open parentheses x close parentheses comma use synthetic division with k equals 2 colon

table attributes columnalign left end attributes row cell table row 2 row blank end table stack open vertical bar table attributes columnalign right end attributes row 1 cell space 3 end cell cell short dash 11 end cell 2 row blank 2 10 cell short dash 2 end cell end table close with bar below end cell row cell space space space space space space 1 space space space 5 space space space space short dash 1 space space space space space 0 end cell end table

Now, write f open parentheses x close parentheses in factored form: f open parentheses x close parentheses equals open parentheses x minus 2 close parentheses open parentheses x squared plus 5 x minus 1 close parentheses

Now, solve open parentheses x minus 2 close parentheses open parentheses x squared plus 5 x minus 1 close parentheses equals 0.

open parentheses x minus 2 close parentheses open parentheses x squared plus 5 x minus 1 close parentheses equals 0 This is the original equation.
x minus 2 equals 0 comma space x squared plus 5 x minus 1 equals 0 The quadratic doesn’t factor; set each factor equal to 0.
x equals 2 comma space x equals fraction numerator short dash 5 plus-or-minus square root of 5 squared minus 4 open parentheses 1 close parentheses open parentheses short dash 1 close parentheses end root over denominator 2 open parentheses 1 close parentheses end fraction Solve the first equation; use the quadratic formula for the second equation.
x equals 2 comma space x equals fraction numerator short dash 5 plus-or-minus square root of 29 over denominator 2 end fraction Simplify.

The zeros of f open parentheses x close parentheses are 2, fraction numerator short dash 5 plus square root of 29 over denominator 2 end fraction comma and fraction numerator short dash 5 minus square root of 29 over denominator 2 end fraction.

try it
Consider the polynomial f open parentheses x close parentheses equals x cubed minus 1.
Given that x - 1 is a factor of f   (x  ), find the zeros of f   (x  ).
First, use division to get the factored form of f open parentheses x close parentheses. In this case, we’ll use synthetic division. The table is below:

table row 1 row blank end table stack open vertical bar table attributes columnalign center right right right end attributes row 1 0 0 cell short dash 1 end cell row blank 1 1 1 end table close with bar below space space space space space space space space space space space 1 space space 1 space space space space space 0

This means that the quotient is x squared plus x plus 1 and f open parentheses x close parentheses equals x cubed minus 1 equals open parentheses x minus 1 close parentheses open parentheses x squared plus x plus 1 close parentheses.

To find the remaining zeros, set x squared plus x plus 1 equals 0 and solve. Noting that this is a quadratic, we’ll use the quadratic formula since factoring does not work.

x squared plus x plus 1 equals 0 The original equation.
x equals fraction numerator short dash open parentheses 1 close parentheses plus-or-minus square root of open parentheses 1 close parentheses squared minus 4 open parentheses 1 close parentheses open parentheses 1 close parentheses end root over denominator 2 open parentheses 1 close parentheses end fraction The quadratic formula with a equals 1 comma b equals 1 comma and c equals 1.
x equals fraction numerator short dash 1 plus-or-minus square root of short dash 3 end root over denominator 2 end fraction Simplify the radicand and the other terms.
x equals fraction numerator short dash 1 plus-or-minus i square root of 3 over denominator 2 end fraction Write the radical in i-form.

Thus, the other zeros are x equals fraction numerator short dash 1 plus-or-minus i square root of 3 over denominator 2 end fraction comma or written individually, x equals fraction numerator short dash 1 plus i square root of 3 over denominator 2 end fraction comma fraction numerator short dash 1 minus i square root of 3 over denominator 2 end fraction.

term to know
Factor Theorem
The number k is a zero of f open parentheses x close parentheses if and only if open parentheses x minus k close parentheses is a factor of f open parentheses x close parentheses.

summary
In this lesson, you learned how to evaluate a polynomial using the remainder theorem, noting that the value of f open parentheses k close parentheses is equal to the remainder when a polynomial f open parentheses x close parentheses is divided by x minus k. As a result, you also learned that when the remainder is equal to 0, k is a zero and the factor theorem tells us x minus k is a factor of f open parentheses x close parentheses. It also tells us if x minus k is a factor of f open parentheses x close parentheses, then f open parentheses k close parentheses equals 0. This is an important start in our quest to analyze the zeros of a polynomial, but there is one more question to answer: How do we proceed when we aren’t given a factor of f open parentheses x close parentheses ? That question is answered in the next tutorial.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

Terms to Know
Factor Theorem

The number k is a zero of f open parentheses x close parentheses if and only if open parentheses x minus k close parentheses is a factor of f open parentheses x close parentheses.

Remainder Theorem

When polynomial f open parentheses x close parentheses is divided by x minus k comma the remainder is f open parentheses k close parentheses.

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