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We multiply two rational numbers by multiplying their numerators and multiplying their denominators. For example,

Instead of reducing the rational number after multiplying, it is often easier to reduce before multiplying. We first factor all terms, then divide out the common factors, then multiply:

When we multiply rational numbers, we use the following definition.

 

Multiplication of Rational Numbers

If and are rational numbers, then

 

We multiply rational expressions in the same way that we multiply rational numbers: Factor all polynomials, divide out the common factors, then multiply the remaining factors.

 

Example 1

Multiplying rational expressions

Find each product of rational expressions.

Solution

a) First factor the coefficients in each numerator and denominator:

Factor.
  Divide out the common factors.
  Quotient rule

Caution

Do not attempt to divide out the x in . This expression cannot be reduced because x is not a factor of both terms in the denominator. Compare this expression to the following:

In Example 2(a) we will multiply a rational expression and a polynomial. For Example 2(b) we will use the rule for factoring the difference of two cubes.

 

Example 2

Multiplying rational expressions

Find each product.

Solution

a) First factor the polynomials completely:

 
  Divide out the common factors.
  Multiply.

b) Note that a - b is a factor of a3 - b3 and b - a occurs in the denominator. We can factor b - a as -1(a - b) to get a common factor: