### Multiplication of Rational Expressions

The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers, so let’s start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately:

**Multiplying Rational Expressions Involving Monomials**

1. Multiply the following:

Cancel common factors from the numerator and denominator. The common factors are 4,

2. Multiply

Rewrite the problem as a product of two fractions:

The common factors are 3 and

**Multiplying Rational Expressions Involving Polynomials**

When multiplying rational expressions involving polynomials, first we need to factor all polynomial expressions as much as we can. Then we follow the same procedure as before.

Multiply

Factor all polynomial expressions as much as possible:

The common factors are

**Multiplying a Rational Expression by a Polynomial**

When we multiply a rational expression by a whole number or a polynomial, we can write the whole number (or polynomial) as a fraction with denominator equal to one. We then proceed the same way as in the previous examples.

Multiply

Rewrite the expression as a product of fractions:

Factor polynomials:

The common factor is

### Example

#### Example 1

Multiply

Factor polynomials:

The common factors are

### Review

Multiply the following rational expressions and reduce the answer to lowest terms.

x32y3⋅2y2x 2xy2⋅4y5x 2xy⋅2y2x3 4y2−1y2−9⋅y−32y−1 6aba2⋅a3b3b2 33a2−5⋅2011a3 2x2+2x−24x2+3x⋅x2+x−6x+4 xx−5⋅x2−8x+15x2−3x 5x2+16x+336x2−25⋅(6x2+5x) x2+7x+10x2−9⋅x2−3x3x2+4x−4 x2+8x+167x2+9x+2⋅7x+2x2+4x

### Review (Answers)

To view the Review answers, open this PDF file and look for section 12.8.