# Rational Expressions

A **rational expression** is an algebraic expression of the form \( \frac{A}{B} \), where \(A \) and \(B \) are polynomials, and \( B \neq 0 \).

Here are a few examples of rational expressions where the denominator is simply \( 1 \): \[ 2x, 2x^2, 2x^2 +1. \] The following are a few examples of rational expressions where the denominator is a constant: \[ \frac{2x}{3}, \frac{2x^2}{5}, \frac{2x^2 +1}{4}. \] Also, the following are a few examples of rational expressions where the denominator contains variables: \[ \frac{1}{x}, \frac{x+1}{x}, \frac{x+1}{x-3}. \]

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## Properties of Rational Expressions

Let \( A \), \( B \), and \( C\) be real numbers or variable expressions, where \(B \neq 0 \) and \(C \neq 0 \).

\( \frac{AC}{BC} = \frac{A}{B} \): You can divide out the top and bottom by a common factor \(C\). This is also known as "canceling" \( C \).

\( \frac{A}{B} = \frac{A \times C}{B \times C} \): You can multiply the top and bottom by a common factor \( C \).

## Simplifying Rational Expressions using the Laws of Exponents

## Simplify

\[ \frac{ 15xy^2 }{ 12y }. \]

We have

\[ \frac{ 15xy^2 }{ 12y } = \frac{ 3 \cdot 5 xy^2 }{ 4 \cdot 3 y } = \frac{ 5 }{ 4 }xy^{2-1} = \frac{ 5 }{ 4 }xy. \ _\square \]

## Simplify

\[ \left( \frac{a^5b^{-3}}{a^3b^8} \right)^2 .\]

We have

\[ \left( \frac{a^5b^{-3}}{a^3b^8} \right)^2 = \left(a^{5-3}b^{-3-8}\right)^2 = \left(a^2b^{-11}\right)^2 = a^4b^{-22} = \frac{a^4}{b^{22}}. \ _\square \]

For more examples applying the laws of exponents, see Simplifying Expressions with Exponents.

## Simplifying Rational Expressions by Factoring

## What is the value of

\[\frac {x^2 - 9}{x + 3} \] at \( x = 10 \)?

Factorizing the numerator of the expression gives \[\frac {x^2 - 9}{x + 3} = \frac{(x-3)(x+3)}{x+3}. \] Canceling out the common factor \( x + 3 \), we get \[ \frac{(x-3)(x+3)}{x+3} = x - 3. \] For \(x = 10\), we get \( x - 3 = 10 - 3 = 7 \). \( _\square \)

## Factor

\[ \frac{3x^3 - 6x^2}{9x^2}. \]

Dividing both the numerator and denominator by a common factor of \(3x^2 \), we get

\[ \frac{3x^3 - 6x^2}{9x^2} = \frac{x - 2}{3}. \ _\square \]

## Factor

\[ \frac{x^2 - x - 2}{x^2 - 2x}. \]

The expression can be factored as

\[ \frac{x^2 - x - 2}{x^2 - 2x} =\frac{(x-2)(x+1)}{x(x-2)}. \]

Canceling out the common factor \( x - 2 \),

\[ \frac{(x-2)(x+1)}{x(x-2)} = \frac{x+1}{x}.\ _\square \]

## Factor

\[ \frac{6x^2 - x - 2}{10x^2 + 3x - 1} .\]

We have

\[ \begin{align} \frac{6x^2 - x - 2}{10x^2 + 3x - 1} &=\frac{ (2x + 1)(3x - 2) }{ (2x + 1)(5x - 1) } \\ &= \frac{ 3x - 2 }{ 5x - 1 }.\ _\square \end{align}\]

## Factor

\[ \frac{x^2 - y^2}{x^3 - y^3}. \]

We have

\[ \frac{x^2 - y^2}{x^3 - y^3} =\frac{ (x - y)(x + y) }{ (x - y)(x^2 + xy + y^2) }. \]

Canceling out the common factor \( x - y \),

\[ \frac{ (x - y)(x + y) }{ (x - y)(x^2 + xy + y^2) } = \frac{x+y}{x^2 + xy + y^2}. \ _\square \]

Next, see the Simplifying Rational Expressions page to learn how to multiply, divide, add, and subtract rational expressions.

**Cite as:**Rational Expressions.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/simplify-fractions/