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# Algebra 2 - Simplifying Rational Expressions

## Introduction

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

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

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

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

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

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

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

Solution: $\frac{2x}{\left({x}^{2}+2x\right)}$$=\frac{2x}{x·\left(x+2\right)}$$=\frac{2}{\left(x+2\right)}$

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

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

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

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

## 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 $\frac{2x}{x-3}$ is not defined for $x=3$.