Chapter 7 Rational Functions - College of the Redwoods

668 Chapter 7 Rational Functions
2x 2 − 3x − 9 = 2x 2 + 3x − 6x − 9
= x(2x + 3) − 3(2x + 3)
= (x − 3)(2x + 3).
The denominator of the first fraction in equation (17) easily factors using the difference
of two squares pattern.
x 2 − 16 = (x + 4)(x − 4)
The numerator of the second fraction in equation (17) is a quadratic trinomial, with
ac = (2)(−12) = −24. The integer pair −3 and 8 have product −24 and sum 5. Hence,
2x 2 + 5x − 12 = 2x 2 − 3x + 8x − 12
= x(2x − 3) + 4(2x − 3)
= (x + 4)(2x − 3).
To factor the denominator of the last fraction in equation (17), first pull the greatest
common factor (in this case x), then complete the factorization using the difference of
two squares pattern.
4x 3 − 9x = x(4x 2 − 9) = x(2x + 3)(2x − 3)
We can now replace each numerator and denominator in equation (17) with its factorization,
then cancel common factors.
− 2x2 − 3x − 9
x 2 − 16
· 2x2 + 5x − 12
4x 3 − 9x
(x − 3)(2x + 3)
= −
(x + 4)(x − 4) ·
= −
(x − 3)(2x + 3)
(x + 4)(x − 4) ·
= − x − 3
x(x − 4)
(x + 4)(2x − 3)
x(2x + 3)(2x − 3)
(x + 4)(2x − 3)
x(2x + 3)(2x − 3)
The last denominator has factors x and x − 4, so x = 0 and x = 4 are restrictions.
In the body of our work, the first fraction’s denominator has factors x + 4 and x − 4.
We’ve seen the factor x − 4 already, so only the factor x + 4 adds a new restriction,
x = −4. Again, in the body of our work, the second fraction’s denominator has factors
x, 2x + 3, and 2x − 3, so we have added restrictions x = 0, x = −3/2, and x = 3/2.
There’s one bit of trickery here that can easily be overlooked. In the body of our
work, the second fraction’s numerator was originally a denominator before we inverted
the fraction. So, we must consider what makes this numerator zero as well. Fortunately,
the factors in this numerator are x + 4 and 2x − 3 and we’ve already considered the
restrictions produced by these factors.
Hence, for all values of x, except the restricted values −4, −3/2, 0, 3/2, and 4, the
left-hand side of
9 + 3x − 2x 2
x 2 − 16
÷ 4x3 − 9x
2x 2 + 5x − 12 = − x − 3
x(x − 4)
(18)
Version: Fall 2007