Unit 11.2 Multiplying and Dividing Rational Expressions
Multiplying Rational Expressions
To multiply rational expressions, follow the same rules as multiplying fractions. It is important to factor both the numerator and denominator first so that it can be simplified fully.
Multiplying Rational Expressions
Multiply [latex]\frac{x^2 - 4}{x^2 + x - 6}[/latex] by [latex]\frac{x + 3}{x - 2}[/latex]
Factor the numerators and denominators of both fractions
First Fraction Numerator
[latex]\ x^2 - 4 = (x - 2)(x + 2)[/latex]
First Fraction Denominator
[latex]\ x^2 + x - 6 = (x + 3)(x - 2)[/latex]
The second fraction is fully factored already so we don’t need to try to factor it further. Rewrite the fraction
[latex]\frac{(x - 2)(x + 2)}{(x + 3)(x - 2)} \cdot \frac{x + 3}{x - 2}[/latex]
Multiply the numerators together and the denominators together
[latex]\frac{(x - 2)(x + 2)(x + 3)}{(x + 3)(x - 2)(x - 2)}[/latex]
Cancel out common terms. You can cancel out whole binomials, but not individual terms as part of the binomial
[latex]\frac{\cancel{(x - 2)}(x + 2)\cancel{(x + 3)}}{\cancel{(x + 3)}\cancel{(x - 2)}(x - 2)}[/latex]
The simplified version is
[latex]\frac{(x + 2)}{(x - 2)}[/latex]
Dividing Rational Expressions
Just like dividing fractions, you multiply by the reciprocal of the second fraction. The reciprocal is the value that you get when you interchange the numerator and denominator of the fraction
Dividing Rational Exponents
Divide [latex]\frac{2x^2 - 8}{x^2 - 4}[/latex] by [latex]\frac{x + 2}{2x}[/latex]
Factor the numerators and denominators of both fractions
First Fraction Numerator
[latex]\ 2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)[/latex]
First Fraction Denominator
[latex]\ x^2 - 4 = (x - 2)(x + 2)[/latex]
The second fraction is fully factored already so we don’t need to try to factor it further. Rewrite the fraction
[latex]\frac{2(x - 2)(x + 2)}{(x - 2)(x + 2)} \div \frac{x + 2}{2x}[/latex]
Find the reciprocal of the second fraction
[latex]\frac{2x}{x+2}[/latex]
Multiply the first fraction by the reciprocal of the second
[latex]\frac{2(x - 2)(x + 2)}{(x - 2)(x + 2)} \cdot \frac{2x}{x+2}[/latex]
Multiply the two fractions together
[latex]\frac{2(x - 2)(x + 2) \cdot 2x}{(x - 2)(x + 2) \cdot (x + 2)}[/latex]
Cancel out common terms
[latex]\frac{2 \cancel{(x - 2)}\cancel{(x + 2)} \cdot 2x}{\cancel{(x - 2)}\cancel{(x + 2)} \cdot (x + 2)}[/latex]
Rewrite the fraction
[latex]\frac{2 \cdot 2x}{x + 2}[/latex]
[latex]\frac{4x}{x + 2}[/latex]
The simplified version is
[latex]\frac{4x}{x + 2}[/latex]