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Int. Alg. Notes
Section 6.1
Page 1 of 4
Section 6.1: Multiplying and Dividing Rational Expressions
Big Idea: A rational expression is the quotient of two polynomials. We study them because they incorporate
the arithmetic operation of division (polynomials only used addition, subtraction, and multiplication). We
multiply and divide rational expressions in the same way we multiply or divide arithmetic fractions…
Big Skill: You should be able to identify the domain of a rational expression, simplify a rational expression by
cancelling common factors, and multiply and divide rational expressions using the familiar rules from
arithmetic.
Examples of rational expressions:
x5
1
x 2  7 x  18
2
2x 1
x3
x 4
The domain of a rational expression is the set of all numbers which can be plugged into the rational
expression. To find the domain of a rational expression, find all values of the variable that cause the
denominator to be zero, and exclude them from the domain.
Practice: find the domain of the above rational expressions.
A rational expression is simplified when all common factors in the numerator and denominator have been
cancelled.
Examples:
12 2  2  3 2  2  3 3



20 2  2  5 2  2  5 5
 x  5   x  3 x  5
x 2  2 x  15  x  5 x  3



2
2 x  3x  9  2 x  3 x  3  2 x  3  x  3 2 x  3
Practice:
1. Simplify
x2  x  6
2 x2  5x  2
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
2. Simplify
y 3  27
2 y2  6 y
3. Simplify
10  2 x
4 x 2  20 x
4. Simplify
3w2  13w  10
2  3w
Section 6.1
Page 2 of 4
a c ac
 
b d bd
a c a d ad
To divide rational expressions, use the arithmetic rule    
(i.e., invert and multiply)
b d b c bc
To multiply rational expressions, use the arithmetic rule
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 6.1
Page 3 of 4
Practice:
1. Simplify
n2  9
n2

2
n  5n  6 6  2n
2. Simplify
a 2  b2
10a  5b
 2
2
10a  10ab 2a  3ab  b2
45 z 4
7y
3. Simplify

5z
21y 2
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 6.1
Page 4 of 4
p3  8
5 p 2  15 p
4. Simplify

p2  4
p2  3 p
m 2  5m
m7

5. Simplify
2m
m 2  6m  7
A rational function is a function of the form
p  x
R  x 
q  x
Where p and q are polynomial functions and q is not the zero of the polynomial. The domain consists of all real
numbers except those for which the denominator q is zero.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.