Chapter 7 Rational Functions - College of the Redwoods

628 Chapter 7 Rational Functions
At this point,
2x − 2x 3
3x 3 + 4x 2 − 3x − 4 = 2x(1 + x)(1 − x)
(x + 1)(x − 1)(3x + 4) .
Because we have 1 − x in the numerator and x − 1 in the denominator, we will factor
out a −1 from 1 − x, and because the order of factors does not affect their product, we
will move the −1 out to the front of the numerator.
2x − 2x 3 2x(1 + x)(−1)(x − 1) −2x(1 + x)(x − 1)
3x 3 + 4x 2 = =
− 3x − 4 (x + 1)(x − 1)(3x + 4) (x + 1)(x − 1)(3x + 4)
We can now cancel common factors.
2x − 2x 3 −2x(1 + x)(x − 1)
3x 3 + 4x 2 =
− 3x − 4 (x + 1)(x − 1)(3x + 4)
−2x(1 + x)(x − 1)
=
(x + 1)(x − 1)(3x + 4)
= −2x
3x + 4
Note that x + 1 is identical to 1 + x and cancels. Thus,
2x − 2x 3
3x 3 + 4x 2 − 3x − 4 =
−2x
3x + 4
for all values of x, provided x ≠ −1, 1, or −4/3. These are the restrictions, values of x
that make denominators equal to zero.
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The Sign Change Rule for Fractions
Let’s look at an alternative approach to the last example. First, let’s share the precept
that every fraction has three signs, one on the numerator, one on the denominator, and
a third on the fraction bar. Thus,
−2
3
has understood signs
Let’s state the sign change rule for fractions.
+ −2
+3 .
The Sign Change Rule for Fractions. Every fraction has three signs, one on
the numerator, one on the denominator, and one on the fraction bar. If you don’t
see an explicit sign, then a plus sign is understood. If you negate any two of these
parts,
• numerator and denominator, or
• numerator and fraction bar, or
• fraction bar and denominator,
then the fraction remains unchanged.
Version: Fall 2007