Multiplication of rational expressions works the same way as multiplication of any other fractions. We multiply the numerators to find the numerator of the product, and then multiply the denominators to find the denominator of the product. Before multiplying, it is helpful to factor the numerators and denominators just as we did when simplifying rational expressions. We are often able to simplify the product of rational expressions.
How To: Given two rational expressions, multiply them
Factor the numerator and denominator.
Multiply the numerators.
Multiply the denominators.
Simplify.
Multiply the rational expressions and show the product in simplest form:
[latex]\begin{array}{cc}\dfrac{\left(x+5\right)\left(x - 1\right)}{3\left(x+6\right)}\cdot \dfrac{\left(2x - 1\right)}{\left(x+5\right)}\hfill & \text{Factor the numerator and denominator}.\hfill \\ \dfrac{\left(x+5\right)\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)\left(x+5\right)}\hfill & \text{Multiply numerators and denominators}.\hfill \\ \dfrac{\cancel{\left(x+5\right)}\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)\cancel{\left(x+5\right)}}\hfill & \text{Cancel common factors to simplify}.\hfill \\ \dfrac{\left(x - 1\right)\left(2x - 1\right)}{3\left(x+6\right)}\hfill & \hfill \end{array}[/latex]
Dividing Rational Expressions
Division of rational expressions works the same way as division of other fractions.
To divide one fraction by another, you multiply the first fraction by the reciprocal (inverse) of the second. This method simplifies the process and avoids direct division:
[latex]\begin{array}\text{ }\dfrac{2x^{2}+x-6}{x^{2}-1}\cdot\dfrac{x^{2}+2x+1}{x^{2}-4} \hfill& \text{Rewrite as the first fraction multiplied by the reciprocal of the second fraction.} \\ \dfrac{\left(2x-3\right)\cancel{\left(x+2\right)}}{\cancel{\left(x+1\right)}\left(x-1\right)}\cdot\dfrac{\cancel{\left(x+1\right)}\left(x+1\right)}{\cancel{\left(x+2\right)}\left(x-2\right)} \hfill& \text{Factor and cancel common factors.} \\ \dfrac{\left(2x-3\right)\left(x+1\right)}{\left(x-1\right)\left(x-2\right)} \hfill& \text{Multiply numerators and denominators.} \\ \dfrac{2x^{2}-x-3}{x^{2}-3x+2} \hfill& \text{Simplify.}\end{array}[/latex]