Lesson 8.1: Simplify Rational Expressions

New Concept

A rational expression is a fraction with polynomials, like $\frac{p(x)}{q(x)}$. We'll learn to simplify these by factoring, find values that make them undefined (when $q(x)=0$), and evaluate them, extending our skills with regular fractions.

What’s next

Next, you'll tackle interactive examples and practice cards to master evaluating and simplifying rational expressions, including those with opposite factors.

Property

A rational expression is an expression of the form $\frac{p(x)}{q(x)}$, where $p$ and $q$ are polynomials and $q \neq 0$.

Examples

• The expression $\frac{5}{21}$ is a simple rational expression where the polynomials are constants.
• The expression $\frac{8a}{b^2}$ is a rational expression containing variables, where $b \neq 0$.
• The expression $\frac{x^2+3x-1}{x-5}$ is a rational expression where the numerator and denominator are polynomials and $x \neq 5$.

Explanation

Think of a rational expression as a fraction where the numerator and denominator are polynomials instead of just numbers. The most important rule is that the denominator can never be zero, as division by zero is undefined.

Property

If the denominator is zero, the rational expression is undefined. To determine the values for which a rational expression is undefined:

1. Set the denominator equal to zero.
2. Solve the equation.

Examples

• For $\frac{8z}{z-4}$, set $z-4=0$. The expression is undefined when $z=4$.
• For $\frac{5x-2}{3x+9}$, set $3x+9=0$. Solving gives $3x=-9$, so the expression is undefined when $x=-3$.
• For $\frac{y+1}{y^2-25}$, set $y^2-25=0$. Factoring gives $(y-5)(y+5)=0$, so the expression is undefined when $y=5$ or $y=-5$.

Explanation

A rational expression is undefined when its denominator equals zero, because division by zero is impossible. Finding these 'forbidden' values for the variable is a crucial first step before performing any other operations.

Property

To evaluate a rational expression, we substitute values of the variables into the expression and simplify. Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator.

Examples

• To evaluate $\frac{4x+1}{x-2}$ for $x=3$: $\frac{4(3)+1}{3-2} = \frac{13}{1} = 13$.
• To evaluate $\frac{y^2-9}{y+2}$ for $y=1$: $\frac{(1)^2-9}{1+2} = \frac{1-9}{3} = \frac{-8}{3}$.
• To evaluate $\frac{a^2-ab}{2b}$ for $a=4, b=2$: $\frac{4^2-4(2)}{2(2)} = \frac{16-8}{4} = \frac{8}{4} = 2$.

Explanation

Evaluating an expression means finding its numerical value. Simply plug the given number for the variable into the expression and perform the arithmetic. If the denominator becomes zero, the expression is undefined for that value.

Property

A rational expression is considered simplified if there are no common factors in its numerator and denominator.

HOW TO Simplify a Rational Expression:

1. Factor the numerator and denominator completely.
2. Simplify by dividing out common factors.

Examples

• To simplify $\frac{4x^2y}{12xy^2}$: Factor to get $\frac{4 \cdot x \cdot x \cdot y}{3 \cdot 4 \cdot x \cdot y \cdot y}$. Cancel common factors to get $\frac{x}{3y}$.
• To simplify $\frac{x^2-4}{x^2+x-6}$: Factor to get $\frac{(x-2)(x+2)}{(x+3)(x-2)}$. Cancel the common factor $(x-2)$ to get $\frac{x+2}{x+3}$.
• To simplify $\frac{5y+10}{y^2-4}$: Factor to get $\frac{5(y+2)}{(y-2)(y+2)}$. Cancel the common factor $(y+2)$ to get $\frac{5}{y-2}$.

Property

The opposite of $a-b$ is $b-a$.

An expression and its opposite divide to $-1$.

Examples

• To simplify $\frac{z-9}{9-z}$: Recognize that the numerator and denominator are opposites. The expression simplifies to $-1$.
• To simplify $\frac{2x-8}{16-x^2}$: Factor to get $\frac{2(x-4)}{(4-x)(4+x)}$. Since $(x-4)$ and $(4-x)$ are opposites, this simplifies to $\frac{2(-1)}{4+x} = -\frac{2}{x+4}$.
• To simplify $\frac{x^2-6x+5}{25-x^2}$: Factor to get $\frac{(x-5)(x-1)}{(5-x)(5+x)}$. This simplifies to $\frac{(-1)(x-1)}{5+x} = -\frac{x-1}{x+5}$.

Explanation

When a factor in the numerator is the exact opposite of a factor in the denominator, like $(x-4)$ and $(4-x)$, they don't just disappear. They cancel each other out and leave behind a $-1$.