Multiplying and Dividing Rational Expressions
Learning Objective(s)
· Multiply and divide rational expressions and simplify.
Introduction
Just as we can multiply and divide fractions, we can multiply and divide rational expressions, or fractions that include polynomials. In fact, we use the same processes for multiplying and dividing rational expressions as we use for multiplying and dividing numeric fractions.
There are two ways to go about multiplying fractions:
We can multiply the numerators and the denominators and then simplify the product:
_{} = _{}
Or we can factor and simplify the fractions before performing the multiplication:
_{} = _{} = _{} = _{} = _{}
The same two approaches can be applied to rational expressions. In the following examples, we'll try both techniques: multiply, then simplify; and simplify, then multiply. An important difference between fractions and rational expressions, though, is that we must identify any values for the variables that would result in division by 0 since this is undefined. These excluded values must be eliminated from the domain, the set of all possible values of the variable.
First, we'll multiply and then simplify:
Example  
Problem  Solve _{} 

 

10a^{3} = 0
a = 0
The domain is all a_{} 0 
 Determine if there are excluded values, values of a which result in 0 as a denominator—14 cannot equal 0, 10a^{3} can  
 _{} 
 Multiply the numerators together, and multiply the denominators together  
 _{} 
 Simplify by finding common factors in the numerator and denominator  
 _{}
_{} 
 Use the common factors to rewrite as multiplication by 1  
 _{} 
 Simplify  
Answer  _{}, a_{} 0 


Okay, that worked. But this time we'll simplify first, then multiply. When using this method, it helps to look for the greatest common factor. We can factor out any common factors, but finding the greatest one will take fewer steps.
Example  
Problem  Solve _{} 

 

10a^{3} = 0
a = 0
The domain is all a_{} 0 
 Determine if there are excluded values, values of a which result in 0 in a denominator—14 cannot equal 0, 10a^{3} can  
 _{} 
 Factor the numerators and denominators. Look for the greatest common factors.  
 _{}
_{} 
 Regroup the fractions to express common factors as multiplication by 1.
 
 _{} 
 Multiply  
Answer  _{} , a_{} 0 
 0 is an excluded value because this value of a makes the denominator equal to 0.  
Both methods produced the same answer.
Some rational expressions contain quadratic expressions and other multiterm polynomials. To multiply these rational expressions, factor the polynomials and then look for common factors. Just take it step by step, like in the example below.
Example  
Problem  Multiply _{} 

 
 _{} (2a – 5) (a + 2) = 0 a = 2.5, 2
a^{2} + 2a = 0 a(a + 2) = 0 a = 2, 0
The domain is all a_{} 2, 0, or 2.5 
 Determine if there are any excluded values. To do this, set the denominators equal to 0 and solve for a.  
 _{} 
 Factor the numerators and denominators  
 _{} 
 Regroup to express rational expressions equivalent to 1  
 _{}
_{} 
 Multiply simplified rational expressions.
This expression can be left with the denominator in factored form or multiplied out.  
Answer  _{}, a_{} 2, 0, or 2.5 

 
Note that in the answer above, we cannot simplify the rational expression any further. It may be tempting to express the 5s in the numerator and denominator as the fraction _{}, but these 5s are terms that are part of a factor, not factors by themselves. They cannot be pulled out of their expressions.
Perform the indicated operation and express the answer as a simplified rational expression:
_{}, y ≠ 0, 2, 8, 2
A) _{} B) _{} C) _{} D) _{}

Dividing Rational Expressions
We've seen that we multiply rational expression like we multiply fractions. It should come as no surprise that we also divide rational expressions the same way we divide numerical fractions. Specifically, to divide rational expressions, multiply the rational expression numerator by the reciprocal of the rational expression denominator.
Let’s begin by recalling division of numerical fractions:
_{}= _{} = _{}
To divide rational expressions, the process is the same. But remember, we need to find the excluded values, the variable values that would make either denominator equal zero. But there's an new wrinkle this time—because we divide by multiplying by the reciprocal of one of the rational expressions, we also need to find the values that would make the numerator of that expression equal zero. Check it out:
Example  
Problem  Divide _{} 

 

15x^{3} = 0
x=0 is an excluded value. 
 Find excluded values. 9 and 27 can never equal 0.
Because 15x^{3} becomes the denominator in the reciprocal of _{}, we must find the values of x that would make that term equal 0.  
 _{} 
 Rewrite division as multiplication by the reciprocal.  
 _{} 
 Factor the numerators and denominators.  
 _{}
_{} 
 Reorder factors and express as multiplication by 1.
Simplify  
Answer  _{} , x_{}0 

 
Example  
Problem  Divide _{} 

 
 (x +2) = 0 x = 2
_{} _{} x = 3 or 2
6x^{4 }= 0 x = 0
0, 2 and 3 are excluded values.  Determine the excluded values that make the denominators & the numerator of the divisor equal to 0.  
 _{}  Rewrite division as multiplication by the reciprocal.  
 _{}  Factor the numerators and denominators.  
 _{}  Reorder factors and express as multiplication by 1.
Simplify  
Answer  _{} , x_{}0, 2, or 3 

 
Notice that once we rewrite the division as multiplication by a reciprocal, we follow the same process we used to multiply rational expressions. We can't forget to mention excluded values—all values of the variable that result in a denominator of 0 at any step in the process must be left out of the domain.
Find the quotient and express as a simplified rational expression:
_{}, y ≠ 2 or 1
A) _{}
B) _{}
C) _{}
D) _{}

Summary
Rational expressions are multiplied and divided the same way numeric fractions are. To multiply, first find the greatest common factors of the numerator and denominator. Next, regroup the factors to make fractions equivalent to one. Then, multiply any remaining factors. To divide, first rewrite the division as multiplication by the reciprocal of the denominator. The steps are then the same as for multiplication.
When expressing a product or quotient, it is important to state the excluded values. These are all values of a variable that would make a denominator equal zero at any step in the calculations.