College Algebra 9th txtbk

SECTION B–2 Partial Fractions A-21
So for all x,
If x 3, then
15 5C
C 3
6x 2 14x 27 A(x 3) 2 B(x 2)(x 3) C(x 2)
If x 2, then
There are no other values of x that will cause terms on the right to drop out. Because any
value of x can be substituted to produce an equation relating A, B, and C, we let x 0 and
obtain
Therefore,
25 25A
A 1
27 9A 6B 2C
27 9 6B 6
B 5
Substitute A 1 and C 3.
6x 2 14x 27
(x 2)(x 3) 1
2 x 2 5
x 3 3
(x 3) 2
x 2 11x 15
MATCHED PROBLEM 2 Decompose into partial fractions: .
(x 1)(x 2) 2
EXAMPLE 3 Nonrepeating Linear and Quadratic Factors
5x 2 8x 5
Decompose into partial fractions: .
(x 2)(x 2 x 1)
SOLUTION
First, we see that the quadratic in the denominator can’t be factored further in the real numbers.
Then, we use parts 1 and 3 from Theorem 3 to write
So for all x,
If x 2, then
5x 2 8x 5
(x 2)(x 2 x 1) A
x 2 Bx C
x 2 x 1
If x 0, then, using A 3, we have
5x 2 8x 5 A(x 2 x 1) (Bx C)(x 2)
If x 1, then, using A 3 and C 1, we have
A(x2 x 1) (Bx C)(x 2)
(x 2)(x 2 x 1)
2 3 (B 1)(1)
B 2
9 3A
A 3
5 3 2C
C 1