# How to Reduce Rational Expressions to the Lowest Terms?

The rational function is defined as a polynomial coefficient that denominator has a degree of at least $$1$$. In this step-by-step guide, you learn how to reduce rational expressions to the lowest terms.

Rational expressions are fractions that have a polynomial in the numerator, denominator, or both. Although rational expressions can seem complex because they contain variables, they can be simplified by using techniques

## A step-by-step guide to reducing rational expressions to the lowest terms

To simplify any rational expression, follow these steps:

1. Factor both the numerator and denominator of the rational expression. Remember to write each expression in standard form.
2. List restricted values.
3. Reduce the expression by canceling out common factors in the numerator and denominator.
4. Rewrite the remaining factors in the numerator and denominator. Note any restricted values not implied by the expression.

### Reducing Rational Expressions to the Lowest Terms – Example 1:

Reduce this rational expression to the lowest terms. $$\frac {x^2-3x+2}{x^2-1}$$

First, factor the numerator and denominator of rational expression:

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

Since division by $$0$$ is undefined, here we see that $$x≠​1$$ and $$x≠-1$$.

Then, cancel common factors:

$$=\frac{(x-2)(x-1)}{(x-1)(x+1)}$$ $$=\frac{(x-2)}{(x+1)}$$

$$\frac {(x-2)}{(x+1)}$$ for $$x≠​1$$

The original expression requires $$x≠±1$$. We do not need to pay attention to $$x ≠-1$$ because this is understood from the expression.

### Reducing Rational Expressions to the Lowest Terms – Example 2:

Reduce this rational expression to the lowest terms. $$\frac{(x^2-4)}{(x^2-2x-8)}$$

First, factor the numerator and denominator of rational expression:

$$\frac{(x^2-4)}{(x^2-2x-8)}$$ $$=\frac{(x-2)(x+2)}{(x+2)(x-4)}$$

Since division by $$0$$ is undefined, here we see that $$x≠​-2$$ and $$x≠4$$.

Then, cancel common factors:

$$=\frac{(x-2)(x+2)}{(x+2)(x-4)}$$ $$=\frac{(x-2)}{(x-4)}$$

$$\frac{(x-2)}{(x-4)}$$ for $$x≠​-2$$

The original expression requires $$x≠-2$$ and $$x≠4$$. We do not need to pay attention to $$x ≠4$$ because this is understood from the expression.

## Exercises for Reducing Rational Expressions to the Lowest Terms

### Reduce rational expressions to the lowest terms.

1. $$\color{blue}{\frac{x^2-2x-15}{x^2+x-6}}$$
2. $$\color{blue}{\frac{3x^2-3x}{3x^3-6x^2+3x}}$$
3. $$\color{blue}{\frac{x^2+7x+12}{x^2-6x-27}}$$
4. $$\color{blue}{\frac{6x^2-19x+3}{4x^2-36}}$$
5. $$\color{blue}{\frac{x^2+2x-8}{x^2+6x-16}}$$
1. $$\color{blue}{\frac{x-5}{x-2}}$$, $$\color{blue}{x≠ -3}$$
2. $$\color{blue}{\frac{1}{x-1}}$$, $$\color{blue}{x≠0}$$
3. $$\color{blue}{\frac{x+4}{x-9}}$$,$$\color{blue}{x≠-3}$$
4. $$\color{blue}{\frac{6x-1}{4(x+3)}}$$, $$\color{blue}{x≠3}$$
5. $$\color{blue}{\frac{x+4}{x+8}}$$, $$\color{blue}{x≠2}$$

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