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:

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).

math equation

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:

math equation


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 math equation is a zero (root) of the function:

f(x) = 2x3 + 3x2 - 8x - 12.


Answer




Find Scholarships Today!


Book mark this page in Del.icio.us, Furl, Digg, StumbleUpon, whatever...


Didn't find what you are looking for? Try search:


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

dumbolf memoTST bola shadow factory mindfields trick-hoops-challenge crystal clear