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) = a0xn + a1xn−1 + a2xn−2 + ... + an

where

a0 ≠ 0 and

n is a positive integer, called the degree of the polynomial.

Example 1

f(x) = 7x5 + 4x3 − 2x2 − 8x + 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) = 3x2 + 5x − 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 (xr) and a remainder R is obtained, then f(r) = R.

Example 3

Use the remainder theorem to find the remainder for Example 1 above, which was divide f(x) = 3x2 + 5x − 8 by (x − 2).

Example 4

By using the remainder theorem, determine the remainder when

3x3x2 − 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 of f(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) = x3 + 2x2 − 5x − 6?

Exercises

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

(2x4 - 10x2 + 30x - 60) ÷ (x + 4)

2. Find the remainder using the Remainder Theorem

(x4 − 5x3 + x2 − 2x + 6) ÷ (x + 4)

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

f(x) = x5 − 2x4 + 3x3 − 6x2 − 4x + 8.

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

f(x) = 2x3 + 3x2 − 8x − 12.