Math 102
4.2 "Remainder and Factor Theorems"
Bibiana Lopez
Crafton Hills College
October 2010
(CHC)
4.2
October 2010
1 / 12
Objectives:
* Use the remainder theorem to evaluate a function for a given value.
* Determine if an expression is a factor of a given polynomial.
* Find linear factors of a polynomial.
(CHC)
4.2
October 2010
2 / 12
Remainder Theorem
Let’s consider the division algorithm when the dividend, f (x ), is
divided by a linear polynomial of the form x c. Then the division
algorithm f (x ) = g (x ) q (x ) + r (x ) , (where f (x ) is the dividend,
g (x ) is the divisor, q (x ) is the quotient, and r (x ) is the remainder)
becomes f (x ) = (x c ) q (x ) + r (x ) . Because the degree of the
remainder, r (x ) , must be less than the degree of the divisor, x c, the
remainder is a constant. Therefore, if we let R represent the remainder, we
have f (x ) = (x c ) q (x ) + R . If we evaluate f at c, we obtain
f (c ) = (c c ) q (c ) + R = 0 q (c ) + R = R . In other words, if a
polynomial is divided by a linear polynomial of the form x c, then the
remainder is the value of the polynomial at c.
(CHC)
4.2
October 2010
3 / 12
Remainder Theorem
Remainder Theorem:
If a polynomial f (x ) is divided by x c,
then the remainder is equal to f (c ) .
(CHC)
4.2
October 2010
4 / 12
Remainder Theorem
Example 1: (Using the remainder theorem)
Find f (c ) (i ) by using synthetic division and the remainder theorem and
(ii ) by evaluating f (c ) directly.
a) f (x ) = x 3 + x 2 2x 4 and c = 1
(CHC)
4.2
October 2010
5 / 12
Remainder Theorem
b) f (x ) = 2x 4 + x 3
(CHC)
4x 2
x + 1 and c = 2
4.2
October 2010
6 / 12
Factor Theorem
A general factor theorem can be formulated by considering the
equation f (x ) = (x c ) q (x ) + R . If x c is a factor of f (x ), then
the remainder R must be zero. Conversely, if R = f (c ) = 0, then
f (x ) = (x c ) q (x ) . In other words, x c is a factor of f (x ).
Factor Theorem:
kA polynomial f (x ) has a factor x
(CHC)
c if and only if f (c ) = 0.k
4.2
October 2010
7 / 12
Factor Theorem
Example 2: (Using the factor theorem)
Use the factor theorem to help answer each question about factors.
a) Is x + 3 a factor of 6x 2 + 13x 15?
(CHC)
4.2
October 2010
8 / 12
Factor Theorem
b) Is x
1 a factor of 3x 3 + 5x 2
(CHC)
x
4.2
2?
October 2010
9 / 12
Factor Theorem
Example 3: (Using the factor theorem)
Use synthetic division to show that g (x ) is a factor of f (x ) , and
complete the factorization of f (x ) .
a) g (x ) = x 1; f (x ) = 3x 3 + 19x 2 38x + 16
(CHC)
4.2
October 2010
10 / 12
Factor Theorem
b) g (x ) = x + 2; f (x ) = x 3 + 7x 2 + 4x
(CHC)
4.2
12
October 2010
11 / 12
Factor Theorem
Example 4: (Using the factor theorem)
Find the values of k that make x 1 a factor of k 2 x 4 + 3kx 2
(CHC)
4.2
4.
October 2010
12 / 12