Download 6.1 Multiplying and DIviding Rational Expressions

Math 152 — Rodriguez
Blitzer — 6.1
Multiplying and Dividing Rational Expressions
I. Rational Expressions and Functions
A.
A rational expression is a polynomial divided by a nonzero polynomial.
x+3
x2 − 4
B. A rational function is a function defined by a formula that is a rational expression.
C. Fractions are rational expressions.
II. Domain of Rational Functions
A. We already learned how to find the domain of functions.
• We learned that the domain of a function is the largest set of real numbers for which
the value of f(x) is a real number.
• The only time we have not gotten a real number as a value is when a denominator
is 0.
• An expression where the denominator is equal to 0 is undefined.
• Rational functions can have variables in the denominator and so have the potential
of being undefined expressions for specific values.
Example: Let f ( x ) =
Find f ( 0 ) .
x−3
.
x−7
Find f ( 3) .
Find f ( 7 ) .
B. The domain of a rational function is the set of all real numbers except
_______________________________________________.
Examples: Find the domain of the following functions.
1) g ( x ) =
3x
2
x + 2x − 35
2)
f ( x) =
( x − 4 )2
3x + x − 10
2
3) h ( x ) =
x+7
x2 − 9
III. Simplifying Rational Expressions
A. To simplify, reduce, or write in lowest terms: cancel out any common factors from the
numerator and denominator. That is, the numerator and denominator can’t contain any
common factor (other than 1 or −1).
B. To simplify:
1) Factor completely the numerator and denominator.
2) Cancel out any common factors that the numerator and denominator share.
NEVER cancel TERMS; only FACTORS.
Examples: Simplify.
1)
IV.
y2 − 9
y 2 − 6y + 9
2)
2−x
2
x + 3x − 10
3)
x 2 − 49y 2
x 2 − 14xy + 49y 2
Multiplying Rational Expressions
A. Multiplying rational expressions is the same as multiplying fractions:
A C
⋅ =
B D
A, B, C, D are poly’s; B ≠ 0, D ≠ 0
B. To multiply rational expressions:
1) ______________ __________________ all numerators and denominators.
2) ________ ____ any common factors that any numerator and denominator share.
3) _________ the remaining numerators; __________the remaining denominators.
1)
k 2 − 15k + 56 k 2 − 6k + 5
⋅
k 2 − 17k + 60 k 2 − 19k + 88
Blitzer — 6.1
2)
x3 + 1
4x
⋅
x 3 − x 2 + x −48x − 48
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V. Dividing Rational Expressions
A. Dividing rational expressions is the same as dividing fractions:
A
C
÷
=
B
D
A, B, C, D are poly’s; B ≠ 0, C ≠ 0, D ≠ 0
B. To divide rational expressions:
1. ______ the first rational expression _______; change division to ____________;
use the reciprocal of the divisor (“flip” the divisor).
2. Now that it’s a multiplication problem, proceed with the multiplication process.
1)
p 2 − 10 p + pq − 10q
p − 10
÷
7 p − 7q
2 p 2 − 2q 2
2)
8x 3 − y 3
4x 2 + 2xy + y 2
÷
2
2
2x + y
4x − y
Perform the indicated operation.
6y + 2
3y 2 + y
÷
1− y
y2 − 1
x 2 − y2
x 2 − 9y 2
⋅
x 2 + 4xy + 3y 2 x 2 − 8xy +15y 2
Note: For hw you have a ‘find and interpret’ problem.
Blitzer — 6.1
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