# 2. The Remainder Theorem and the Factor Theorem

This section discusses the historical method of solving higher degree polynomial equations.

As we discussed in the previous section Polynomial Functions and Equations, a **polynomial function** is of the form:

f(x) =a_{0}x+^{n}a_{1}x^{n}^{−1}+a_{2}x^{n}^{−2}+ ... +a_{n}

where

a_{0}≠ 0 and

nis a positive integer, called thedegreeof the polynomial.

**Example 1 **

*f*(*x*) = 7*x*^{5} + 4*x*^{3} − 2*x*^{2} − 8*x* + 1 is a polynomial function of degree 5.

## Dividing Polynomials

First, let's consider what happens when we divide numbers.

**Example (a):** Say we try to divide `13` by `5`. We will get the answer `2` and have a remainder of `3`. We could write this as:

`13/5 = 2 + 3/5`

Another way of thinking about this example is:

`13 = 2 × 5 + 3`

**Example (b), Long Division:** In primary school, you may have learned to divide larger numbers as follows. Let's divide `3,756` by `23`.

`163` | |||

`23` | `{:)` | `3756` | |

`23` | We multiply `23` by `1 = 23`. | ||

`145` | `37-23 = 14`. Then bring down the `5`. | ||

`138` | Multiply `23` by `6=138`. | ||

`76` | `145-138=7`. Bring down the `6`. | ||

`69` | Multiply `23` by `3=69`. | ||

`7` | `76-69=7`. This is the remainder. |

So we can conclude `3,756 -: 23 = 163 + 7/23`, or putting it another way, `3,756 = 163xx23 + 7`.

**Division of polynomials** is an extension of our number examples.

If we divide a polynomial by (*x *−* r*), we obtain a result of the form:

f(x) = (x−r)q(x) +R

where *q*(*x*) is the quotient and *R* is the remainder.

Let's now see an example of polynomial division.

**Example 2 **

Divide *f*(*x*) = 3*x*^{2} + 5*x *− 8 by (*x* − 2).

## The Remainder Theorem

Consider *f*(*x*) = (*x *−* r*)*q*(*x*) +* R*

Note that if we let *x* = *r*, the expression becomes

f(r) = (r−r)q(r) +R

Simplifying gives:

f(r) =R

This leads us to the **Remainder Theorem** which states:

If a polynomial

f(x) is divided by (x−r) and a remainderRis obtained, thenf(r) =R.

### Example 3

Use the remainder theorem to find the remainder for Example 1 above, which was divide *f*(*x*) = 3*x*^{2} + 5*x *− 8 by (*x* − 2).

**Example 4 **

By using the remainder theorem, determine the remainder when

3

x^{3}−x^{2}− 20x+ 5

is divided by (*x* + 4).

## The Factor Theorem

The Factor Theorem states:

If the remainder

f(r) =R= 0, then (x−r) is a factor off(x).

The Factor Theorem is powerful because it can be used to find roots of polynomial equations.

**Example 5 **

Is (*x *+ 1) a factor of *f*(*x*) =* x*^{3} + 2*x*^{2} − 5*x *− 6?

### Exercises

**1.** Find the remainder *R* by long division **and** by the Remainder Theorem.

(2

x^{4}- 10x^{2}+ 30x- 60) ÷ (x+ 4)

**2.** Find the remainder using the Remainder Theorem

(

x^{4}− 5x^{3}+x^{2}− 2x+ 6) ÷ (x+ 4)

**3.** Use the Factor Theorem to decide if (*x* − 2) is a factor of

f(x) =x^{5}− 2x^{4}+ 3x^{3}− 6x^{2}− 4x+ 8.

**4.** Determine whether `-3/2` is a zero (root) of the function:

f(x) = 2x^{3}+ 3x^{2}− 8x− 12.

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