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Math 102
4.2 "Remainder and Factor Theorems"
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.
Remainder Theorem
Let’s consider the division algorithm when the dividend, f (x), is divided by a linear polynomial of the form x
the division algorithm f (x) = g (x) q (x) + r (x)
and r (x) is the remainder) becomes f (x) = (x
than the degree of the divisor, x
f (x) = (x
c) q (x) + R
:
;
c. Then
(where f (x) is the dividend, g (x) is the divisor, q (x) is the quotient,
c) q (x) + r (x)
:
Because the degree of the remainder, r (x) ; must be less
c; the remainder is a constant. Therefore, if we let R represent the remainder, we have
If we evaluate f at c, we obtain
:
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.
Remainder Theorem:
kIf a polynomial f (x) is divided by x
c, then the remainder is equal to f (c) :k
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.
3
4
a) f (x) = x + x
2
2x
4 and c =
1
b) f (x) = 2x + x3
4x2
x + 1 and c = 2
Factor Theorem
A general factor theorem can be formulated by considering the equation f (x) = (x
of f (x), then the remainder R must be zero. Conversely, if R = f (c) = 0; then f (x) = (x
c) q (x) + R
:
If x
c is a factor
c) q (x) : In other words, x
c
is a factor of f (x).
Factor Theorem:
kA polynomial f (x) has a factor x
c if and only if f (c) = 0:k
Page: 1
Notes by Bibiana Lopez
College Algebra by Kaufmann and Schwitters
4.2
Example 2: (Using the factor theorem)
Use the factor theorem to help answer each question about factors.
a) Is x + 3 a factor of 6x2 + 13x
15?
b) Is x
1 a factor of 3x3 + 5x2
x
2?
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) = 3x3 + 19x2
b) g (x) = x + 2; f (x) = x3 + 7x2 + 4x
38x + 16
12
Example 4: (Using the factor theorem)
Find the values of k that make x
1 a factor of k 2 x4 + 3kx2
4.
Page: 2
Notes by Bibiana Lopez