# Simplify Rational Expressions

 How to simplify rational expressions? A tutorial with examples and detailled solutions is presented. To simplify a rational expression, we first factor both the numerator and denominator completely then reduce the expression by cancelling common factors. Example 1: Simplify the rational expression $\dfrac{4x-2}{2x-1}$ Detailed Solution to Example 1 Factor both the numerator and denominator completely. $\dfrac{2(2x - 1)}{2x-1}$ Cancel common factors to reduce and simplify the given expression. $\dfrac{2\cancel{(2x-1)}}{\cancel{2x-1}}=2$ , with $x$ not equal to $1$. Note: The given expression is equal to 2 on the condition that $x$ is not equal to $1$. $x = 1$ is not in the domain of the expression. Example 2: Simplify the rational expression $\dfrac{4 x+16}{x^2-16}$ Detailed Solution to Example 2 Factor both the numerator and denominator completely. $\dfrac{4 x+16}{x^2-16}=\dfrac{4(x+4)}{(x+4)(x-4)}$ Cancel common factors to simplify the expression. $=\dfrac{4\cancel{(x+4)}}{(x-4)\cancel{(x+4)}}$ $=\dfrac{4}{x-4}$ , with $x$ not equal to $-4$. Example 3: Simplify the rational expression $\dfrac{x^2+x-2}{-x^2-2x+3}$ Detailed Solution to Example 3 Factor both the numerator and denominator completely. $\dfrac{x^2+x-2}{-x^2-2x+3}=\dfrac{(x+2)(x-1)}{(x+3)(-x+1)}$ Note that $- x + 1 = - (x - 1)$ in the denominator. $\dfrac{(x+2)(x-1)}{-(x+3)(x-1)}$ Cancel common factors. $=\dfrac{(x+2)\cancel{(x-1)}}{-(x+3)\cancel{(x-1)}}$ $=-\dfrac{x+2}{x+3}$ , with $x$ not equal to $1$. Exercises Simplify the rational expression. $\dfrac{4x+2}{(x-9)(2x+1)}$ $\dfrac{x^2 + 7x + 12}{-x^2 - 5x - 6}$ $\dfrac{x^4-1}{(x^2+ 1)(x - 1)}$ $\dfrac{x^3 + x^2 - 2x}{x-1}$ More references to topics related to rational expressions. Definition and domain of Rational Expressions.

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Updated: 2 April 2013