# 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

## Related Topics

- How to Add and Subtract Rational Expressions
- How to Multiply Rational Expressions
- How to Graph Rational Functions

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

To simplify any rational expression, follow these steps:

- Factor both the numerator and denominator of the rational expression. Remember to write each expression in standard form.
- List restricted values.
- Reduce the expression by canceling out common factors in the numerator and denominator.
- 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:

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

Now, write the final answer:

\(\frac {(x-2)}{(x+1)}\) 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:

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

Now, write the final answer:

\(\frac{(x-2)}{(x-4)}\) 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.

- \(\color{blue}{\frac{x^2-2x-15}{x^2+x-6}}\)
- \(\color{blue}{\frac{3x^2-3x}{3x^3-6x^2+3x}}\)
- \(\color{blue}{\frac{x^2+7x+12}{x^2-6x-27}}\)
- \(\color{blue}{\frac{6x^2-19x+3}{4x^2-36}}\)
- \(\color{blue}{\frac{x^2+2x-8}{x^2+6x-16}}\)

- \(\color{blue}{\frac{x-5}{x-2}}\), \(\color{blue}{x≠ -3}\)
- \(\color{blue}{\frac{1}{x-1}}\), \(\color{blue}{x≠0}\)
- \(\color{blue}{\frac{x+4}{x-9}}\),\(\color{blue}{x≠-3}\)
- \(\color{blue}{\frac{6x-1}{4(x+3)}}\), \(\color{blue}{x≠3}\)
- \(\color{blue}{\frac{x+4}{x+8}}\), \(\color{blue}{x≠2}\)

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