Chapter Contents

• Polynomial Equations of Higher Degree
• 1. Polynomial Functions and Equations
• 2. Remainder and Factor Theorems
• 3. Factors and Roots of Polynomial Equations
• Polynomial Equations of Higher Degree Problem Solver

• Comments, Questions?

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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 + a₁x^n-1 + a₂x^n-2 + ... + a_n

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.

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`

Division of polynomials is something like our number example.

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.

Example 2

Divide f(x) = 3x² + 5x − 8 by (x − 2).

Answer

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 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).

Answer

Example 4

By using the remainder theorem, determine the remainder when

3x³ − x² − 20x + 5

is divided by (x + 4).

Answer

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?

Answer

Exercises

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

(2x⁴ - 10x² + 30x - 60) ÷ (x + 4)

Answer

2. Find the remainder using the Remainder Theorem

(x⁴ − 5x³ + x² − 2x + 6) ÷ (x + 4)

Answer

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

f(x) = x⁵ − 2x⁴ + 3x³ − 6x² − 4x + 8.

Answer

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

f(x) = 2x³ + 3x² − 8x − 12.

Answer