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# 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 $\frac{25{x}^{5}}{8{y}^{2}}×\frac{72{y}^{3}}{100{x}^{7}}$

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

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

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

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

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

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

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

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

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

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

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

$=\frac{\left(3x-1\right)\left(x+1\right)}{3{x}^{2}}×\frac{2x}{\left(x+1\right)\left(x-2\right)}$

$=\frac{\left(3x-1\right)\left(x+1\right)×2x}{3{x}^{2}×\left(x+1\right)\left(x-2\right)}$

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

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

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

$=\frac{2\left(x-3\right)\left(x-1\right)}{\left(x+2\right)\left(2x-1\right)}÷\frac{\left(x-3\right)\left(x+1\right)}{\left(2x-1\right)}×\frac{\left(x+1\right)\left(x+2\right)}{\left(x-3\right)}$

$=\frac{2\left(x-3\right)\left(x-1\right)}{\left(x+2\right)\left(2x-1\right)}×\frac{\left(2x-1\right)}{\left(x-3\right)\left(x+1\right)}×\frac{\left(x+1\right)\left(x+2\right)}{\left(x-3\right)}$

$=\frac{2\left(x-3\right)\left(x-1\right)×\left(2x-1\right)×\left(x+1\right)\left(x+2\right)}{\left(x+2\right)\left(2x-1\right)×\left(x-3\right)\left(x+1\right)×\left(x-3\right)}$

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

## 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.