Chapter 7 Rational Functions - College of the Redwoods

Section 7.4 Products and Quotients of Rational Functions 667
12 ÷ 1 2
= 12 · 2 = 24
This makes sense, as there are 24 “halves” in 12. Let’s look at a harder example.
◮ Example 13.
Simplify
33
15 ÷ 14
10 . (14)
Invert the second fraction and multiply. After that, all we need to do is factor
numerators and denominators, then cancel common factors.
33
15 ÷ 14
10 = 33
15 · 10
14 = 3 · 11
3 · 5 · 2 · 5
2 · 7 = 3 · 11
3 · 5 · 2 · 5
2 · 7 = 11
7
An interesting way to check this result on your calculator is shown in the sequence of
screens in Figure 3.
(a) (b) (c)
Figure 3.
Using the calculator to check division of fractions.
After entering the original problem in your calculator, press ENTER, then press the MATH
button, then select 1:◮ Frac from the menu and press ENTER. The result is shown in
Figure 3(c), which agrees with our calculation above.
Let’s look at another example.
◮ Example 15.
State the restrictions.
Simplify
9 + 3x − 2x 2
x 2 − 16
÷ 4x3 − 9x
2x 2 + 5x − 12 . (16)
Note the order of the first numerator differs from the other numerators and denominators,
so we “anticipate” the need for a sign change, negating the numerator and
fraction bar of the first fraction. We also invert the second fraction and change the
division to multiplication (“invert and multiply”).
− 2x2 − 3x − 9
x 2 − 16
· 2x2 + 5x − 12
4x 3 − 9x
The numerator in the first fraction in equation (17) is a quadratic trinomial, with
ac = (2)(−9) = −18. The integer pair 3 and −6 has product −18 and sum −3. Hence,
(17)
Version: Fall 2007