Algebra 2 - Simplifying Rational Expressions

Introduction

Just like rational numbers (which are expressed in the form of $\frac{p}{q}$ , where $p$ and $q$ are integers and $q \neq 0$ ), rational expressions are simply ratios of two polynomials.
Mathematically, a rational expression in variable ' $x$ ' can be represented as $\frac{f (x)}{g (x)}$ , in which the value of $g (x)$ can never be zero.
Some examples of rational expressions include: $\frac{2 x^{2} + 4 x + 1}{7 x - 5}$ , $\frac{4}{1 - x}$ , $2 x^{3} - 5 x + 3$ , etc.

Simplifying Rational Expressions

In general, simplifying any rational expression can be accomplished in two steps.

1. Factorize the polynomials of the numerator and denominator
2. Reduce the expression by canceling out common factors

Solved Examples

Example 1: Is $\frac{1 + \sqrt{x}}{2 x - 5}$ a rational expression?

Answer: No, because the numerator is not a polynomial.

Example 2: Is $2 x - 9$ as rational expression?

Solution: Yes, because $2 x - 9$ can be expressed as $\frac{2 x - 9}{1}$ which satisfied the definition of a rational expression.

Example 3: Simplify $\frac{x^{2} - 16}{x + 4}$ .

Solution: $\frac{x^{2} - 16}{x + 4}$ $= \frac{(x + 4) \cdot (x - 4)}{(x + 4)}$ $= (x - 4)$

Example 4: Simplify $\frac{2 x}{(x^{2} + 2 x)}$ .

Solution: $\frac{2 x}{(x^{2} + 2 x)}$ $= \frac{2 x}{x \cdot (x + 2)}$ $= \frac{2}{(x + 2)}$

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

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

Example 6: Simplify $\frac{x^{2} + 4 x - 5}{x^{2} - 5 x + 4}$ .

Solution: $\frac{x^{2} + 4 x - 5}{x^{2} - 5 x + 4}$ $= \frac{(x - 1) (x + 5)}{(x - 1) (x - 4)}$ $= \frac{(x + 5)}{(x - 4)}$

Cheat Sheet

To simplify any rational expression, factorize the polynomials in the numerator and denominator and cancel out the common factors.

Blunder Areas

A rational expression $\frac{2 x}{x - 3}$ is not defined for $x = 3$ .