12.5: Multiplying and Dividing Rational Expressions
Learning Objectives
 Multiply rational expressions involving monomials.
 Multiply rational expressions involving polynomials.
 Multiply a rational expression by a polynomial.
 Divide rational expressions involving polynomials.
 Divide a rational expression by a polynomial.
 Solve realworld problems involving multiplication and division of rational expressions.
Introduction
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:
When we divide two fractions, we replace the second fraction with its reciprocal and multiply, since that’s mathematically the same operation:
Multiply Rational Expressions Involving Monomials
Example 1
Multiply the following:
Solution
Cancel common factors from the numerator and denominator. The common factors are 4,
Example 2
Multiply
Solution
Rewrite the problem as a product of two fractions:
The common factors are 3 and
Multiply 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.
Example 3
Multiply
Solution
Factor all polynomial expressions as much as possible:
The common factors are
Example 4
Multiply
Solution
Factor polynomials:
The common factors are
Multiply 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.
Example 5
Multiply
Solution
Rewrite the expression as a product of fractions:
Factor polynomials:
The common factor is
Divide Rational Expressions Involving Polynomials
Just as with ordinary fractions, we first rewrite the division problem as a multiplication problem and then proceed with the multiplication as outlined in the previous example.
Note: Remember that
Example 6
Divide
Solution
First convert into a multiplication problem by flipping the second fraction and then simplify as usual:
Example 7
Divide
Solution
Divide a Rational Expression by a Polynomial
When we divide 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, and then proceed the same way as in the previous examples.
Example 8
Divide
Solution
Rewrite the expression as a division of fractions, and then convert into a multiplication problem by taking the reciprocal of the divisor:
Then factor and solve:
For more examples of how to multiply and divide rational expressions, watch the video at
.
Solve RealWorld Problems Involving Multiplication and Division of Rational Expressions
Example 9
Suppose Marciel is training for a running race. Marciel’s speed (in miles per hour) of his training run each morning is given by the function
Solution
Marciel will run for
Review Questions
Perform the indicated operation and reduce the answer to lowest terms.

x32y3⋅2y2x 
2xy÷2x2y 
2xy2⋅4y5x 
2x3y÷3x2 
2xy⋅2y2x3 
3x+6y−4÷3y+9x−1 
4y2−1y2−9⋅y−32y−1 
6aba2⋅a3b3b2 
x2x−1÷xx2+x−2 
33a2−5⋅2011a3 
a2+2ab+b2ab2−a2b÷(a+b)  \begin{align*}\frac{2x^2+2x24}{x^2+3x} \cdot \frac{x^2+x6}{x+4}\end{align*}
 \begin{align*}\frac{3x}{3x5} \div \frac{x^29}{2x^28x10}\end{align*}
 \begin{align*}\frac{x^225}{x+3} \div (x5)\end{align*}
 \begin{align*}\frac{2x+1}{2x1} \div \frac{4x^21}{12x}\end{align*}
 \begin{align*}\frac{x}{x5} \cdot \frac{x^28x+15}{x^23x}\end{align*}
 \begin{align*}\frac{3x^2+5x12}{x^29} \div \frac{3x4}{3x+4}\end{align*}
 \begin{align*}\frac{5x^2+16x+3}{36x^225} \cdot (6x^2+5x)\end{align*}
 \begin{align*}\frac{x^2+7x+10}{x^29} \cdot \frac{x^23x}{3x^2+4x4}\end{align*}
 \begin{align*}\frac{x^2+x12}{x^2+4x+4} \div \frac{x3}{x+2}\end{align*}
 \begin{align*}\frac{x^416}{x^29} \div \frac{x^2+4}{x^2+6x+9}\end{align*}
 \begin{align*}\frac{x^2+8x+16}{7x^2+9x+2} \cdot \frac{7x+2}{x^2+4x}\end{align*}
 Maria’s recipe asks for \begin{align*}2 \frac{1}{2}\end{align*} times more flour than sugar. How many cups of flour should she mix in if she uses \begin{align*}3 \frac{1}{3}\end{align*} cups of sugar?
 George drives from San Diego to Los Angeles. On the return trip he increases his driving speed by 15 miles per hour. In terms of his initial speed, by what factor is the driving time decreased on the return trip?
 Ohm’s Law states that in an electrical circuit \begin{align*}I = \frac{V}{R_c}\end{align*}. The total resistance for resistors placed in parallel is given by: \begin{align*}\frac{1}{R_{tot}} = \frac{1}{R_1} + \frac{1}{R_2}\end{align*}. Write the formula for the electric current in term of the component resistances: \begin{align*}R_1\end{align*} and \begin{align*}R_2\end{align*}.
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