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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:

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 1

Divide f(x)= 3x²+ 5x - 8 by (x − 2).

Thus, we can conclude that:

3x² + 5x − 8 = (x − 2)(3x + 11) + 14

where the quotient q(x) = 3x + 11 and the remainder R = 14.

We can also write our answer as:

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 2

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 3

By using the remainder theorem, determine the remainder when

3x³ − x² − 20x + 5

is divided by (x + 4).

Answer

The Factor Theorem

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 4

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 is a zero (root) of the function:

f(x) = 2x³ + 3x² - 8x - 12.

Answer

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

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

Following are the original SNB files (.tex or .rap) used in making this chapter. For more information, go to SNB info.

SNB files

1. Polynomial Functions and Equations

3. Factors and Roots of Polynomial Equations

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