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: \[ \frac{1}{x}, \frac{x+1}{x}, \frac{x+1}{x-3}. \]
Contents
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.