Download A2 9-1 Notes Teacher Edition

A2
9-1: Multiplying and Dividing Rational Expressions
Notes
Vocabulary
page
472
term
definition
rational
expression
a ratio of two polynomial expressions, of the form p(x) ./. q(x)
ratio
division, quotient
polynomial
of
degree n
an expression of the form
472
excluded
values
the values of the variable for which the expression is undefined
222
mononomial an expression that is a number, a variable, or the product of a number and
one or more variables
475
complex
fraction
346
a0 xn + a1 xn-1 + ... + an-2 x2 + an-1 x + an
The a's are coefficients, and real numbers; a0 is not 0.
The n's are exponents; they are non-negative integers.
The degree is n, the highest exponent.
a rarional expression of which the numerator and/or the denominator
contains a rational expression
There are two things we need to start with in rational expressions: 1) simplify, and 2) determine when
the rational expression is undefined.
1) Simplify
How do you simplify fractions? ___By looking for common factors______________
10 2.5 2
=
=
15 3.5 3
For rational expressions, you do the same: ___you look for common factors______________
Example 1:
Simplify
2x x− 5
x− 5 x 2 − 1
(p.472)
Sometimes it is hard to find common factors. Just remember that
(a - b) = (-1) (b - a)
2
2
z w− z
Example 2: Simplify
z 3− z 3 w
2) Determine when the expression is undefined
2
=?
0
(p. 473)
When is a fraction undefined?___when the denominator equals 0________________
To find out when a rational expression is undefined, you have to find when the denominator equals 0.
That means that you have to completely factor the original denominator. (Why?) Because the
simplified denominator will miss (common) factors.
Example 3:
2x x− 5
x− 5 x 2 − 1
Find the excluded values
(p.473)
Multiplying rational expressions works just like multiplying fractions.
2 1
. =¿
3 4
First you multiply the numerators, and then you multiply the denominators.
2
Example 4:
Simplify
4a 15b
.
5b 16a 3
(p. 474)
Dividing rational expressions works just like dividing fractions.
3 9
/ =¿
7 14
You multiply the numerator by the reciprocal of the denominator.
2
Example 5:
Simplify
2
4x y 2x y
÷
3 3
3
15 a b 5a b
(p. 474)
So far , we have worked with mononomials in our rational expressions. But you follow the same
method if there are polynomials in the numerator or the denominator.
2
Example 6:
Simplify
x 2x− 8 3x 3
.
x 2 4x 3 x − 2
(p. 475)
Complex fractions have rational expressions in their rational expressions. You handle them just like a
number fraction:
2
9
=¿
1
3
Example 7:
Simplify
r2
r 2− 25s 2
r
5s− r
(p. 475)