Section 2.6: Multiplying and Dividing Rational
expressions
Expectations:
1. Apply the steps below to factor and simplify the
product and quotient of rational expressions
Strategy for simplifying the product/quotient of
rational expressions
Step 1: Factor both the numerator and the
denominator.
Step 2 (Optional): Write as one fraction.
Write it as a product of the factors of the
numerators over the product of the factors of the
denominators. DO NOT multiply anything out at
this point.
Step 3: Simplify the rational expression.
Cancel equivalent factors.
Step 4: Multiply any remaining factors in the
numerator and/or denominator.
Example 1: Multiply .
Step 1: Factor both the numerator and the denominator AND
Step 2: Write as one fraction.
*Factor the num. and den.
In the numerator we factored a
difference of squares.
In the denominator we factored a GCF
and a trinomial.
Step 3: Simplify the rational expression.
Step 4: Multiply any remaining factors in the numerator and/or
denominator.
*Simplify by
div. out the common factors
of (y + 3), (y - 3) and y
*Excluded values of the original den.
Also note that the values that would be
excluded from the domain are 0, 3, -6,
and -3. Those are the values that make
the original denominator equal to 0.
Example 2: Multiply .
Step 1: Factor both the numerator and the denominator
Step 2: Write as one fraction
*Factor the num. and den.
In the numerator we factored
a difference of cubes and a
GCF.
In the denominator we
factored a trinomial.
Step 3: Simplify the rational expression.
Step 4: Multiply any remaining factors in the numerator and/or
denominator.
*Simplify by div. out the
common factors of
(x - 3), 2, and (x + 2).
Note that the values that
would be excluded from the
domain are 0, 3, and -2.
Those are the values that
makes the original
denominator equal to 0.
Dividing Rational Expressions
where Q, S, and R do not equal 0.
Step 1: Write as multiplication of the reciprocal.
Step 2: Multiply the rational expressions as shown above.
Example 3: Divide
Step 1: Write as multiplication of the reciprocal Step 2:
Multiply the rational expressions as shown above.
Rewrite as mult. of
reciprocal
Factor the num. and den.
Simplify by div. out the
common factors of 3x and
(x + 6)
Multiply the den.
In the numerator of the
product we factored a
GCF.
In the denominator we factored a trinomial.
Note that the values that would be excluded from the domain are -6 and 0. Those are
the values that makes the original denominator of the product equal to 0.
Example 4: Divide
.
Step 1: Write as multiplication of the reciprocal
AND
Step 2: Multiply the rational expressions as shown above.
*Rewrite as mult. of reciprocal
*Factor the num. and den.
*Simplifyby div. out the common factors of
y, (y + 4), and (y - 4)
*Multiply the num. and den. out
*Excluded values of the original den. of quotient &
product
In the numerator of the product we factored a GCF and a trinomial.
In the denominator we factored a GCF and a difference of
squares.
Note that the values that would be excluded from the domain
are 0, 2, - 4, 4, and -3. Those are the values that make the
original denominator of the quotient and the product equal to
0.