Algebra 2 - Multiplying and Dividing Rational Expressions

Introduction

Multiplication of rational expressions is just like multiplying fractions.
When two rational expressions are multiplied, their respective numerators and denominators are multiplied together and further simplified to yield the final result.
Division of rational expressions is also performed just like dividing fractions.
Division of two rational expressions is equivalent to multiplying the first rational expression by the reciprocal of the second rational expression.

Multiplying Rational Expressions

The steps involved in the multiplication of rational expressions are summarized below.

Factor the polynomials of the numerator and denominator.
Simplify (cancel out common factors) the rational expression.
Finally, multiply the remaining terms by numerator and denominator.

Dividing Rational Expressions

Division of rational expressions can be carried out in the following steps:

Express the division of two rational expressions as multiplication of the first rational expression by the reciprocal of the second rational expression.
Factor the polynomials of the numerator and denominator.
Simplify (cancel out the common factors) the rational expression.
Multiply the terms left in the numerator and denominator.

Solved Examples

Example 1: Simplify $\frac{25 x^{5}}{8 y^{2}} \times \frac{72 y^{3}}{100 x^{7}}$ .

Solution: $\frac{25 x^{5}}{8 y^{2}} \times \frac{72 y^{3}}{100 x^{7}}$ $= \frac{25 \times 72 \times x^{5} y^{3}}{8 \times 100 \times x^{7} y^{2}}$ $= \frac{9 y}{4 x^{2}}$

Example 2: Simplify $\frac{36}{18 x + 9 y} \times \frac{22 x + 11 y}{121}$ .

Solution: $\frac{36}{18 x + 9 y} \times \frac{22 x + 11 y}{121}$ $= \frac{36}{9 (2 x + y)} \times \frac{11 (2 x + y)}{121}$

$= \frac{36 \times 11 \times (2 x + y)}{9 \times 121 \times (2 x + y)}$ $= \frac{4}{11}$

Example 3: Simplify $\frac{2 x^{2} - 7 x + 3}{2 x} \times \frac{18 x^{3}}{2 x^{2} + 5 x - 3}$

Solution: $\frac{2 x^{2} - 7 x + 3}{2 x} \times \frac{18 x^{3}}{2 x^{2} + 5 x - 3}$ $= \frac{(x - 3) (2 x - 1)}{2 x} \times \frac{18 x^{3}}{(2 x - 1) (x + 3)}$

$= \frac{18 x^{3} \cdot (x - 3) (2 x - 1)}{2 x \cdot (2 x - 1) (x + 3)}$ $= \frac{9 x^{2} \cdot (x - 3)}{(x + 3)}$

Example 4: Simplify $\frac{81}{x^{2}} \div \frac{18}{x^{5}}$ .

Solution: $\frac{81}{x^{2}} \div \frac{18}{x^{5}}$ $= \frac{81}{x^{2}} \times \frac{x^{5}}{18}$ $= \frac{81 x^{5}}{18 x^{2}}$ $= \frac{9 x^{3}}{2}$

Example 5: Simplify $\frac{3 x^{2} + 2 x - 1}{3 x^{2}} \div \frac{x^{2} - x - 2}{2 x}$ .

Solution: $\frac{3 x^{2} + 2 x - 1}{3 x^{2}} \div \frac{x^{2} - x - 2}{2 x}$

$= \frac{(3 x - 1) (x + 1)}{3 x^{2}} \times \frac{2 x}{(x + 1) (x - 2)}$

$= \frac{(3 x - 1) (x + 1) \times 2 x}{3 x^{2} \times (x + 1) (x - 2)}$

$= \frac{2 \cdot (3 x - 1)}{3 x \cdot (x - 2)}$

Example 6: Simplify $\frac{2 x^{2} - 8 x + 6}{2 x^{2} + 3 x - 1} \div \frac{x^{2} - 2 x - 3}{2 x - 1} \times \frac{x^{2} + 3 x + 2}{x - 3}$

Solution: $\frac{2 x^{2} - 8 x + 6}{2 x^{2} + 3 x - 1} \div \frac{x^{2} - 2 x - 3}{2 x - 1} \times \frac{x^{2} + 3 x + 2}{x - 3}$

$= \frac{2 (x - 3) (x - 1)}{(x + 2) (2 x - 1)} \div \frac{(x - 3) (x + 1)}{(2 x - 1)} \times \frac{(x + 1) (x + 2)}{(x - 3)}$

$= \frac{2 (x - 3) (x - 1)}{(x + 2) (2 x - 1)} \times \frac{(2 x - 1)}{(x - 3) (x + 1)} \times \frac{(x + 1) (x + 2)}{(x - 3)}$

$= \frac{2 (x - 3) (x - 1) \times (2 x - 1) \times (x + 1) (x + 2)}{(x + 2) (2 x - 1) \times (x - 3) (x + 1) \times (x - 3)}$

$= \frac{2 (x - 1)}{(x - 3)}$

Cheat Sheet

To multiply two rational expressions, multiply their numerators and then multiply their denominators and simplify it.
To divide two rational expressions, multiply the first rational expression by the reciprocal of the second one and simplify it.

Blunder Areas

First, work on division in complex problems involving combined multiplication and division of rational expressions.

Abhishek Tiwari
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