Lesson 1: Algebraic Fractions

New Concept

Welcome to algebraic fractions, which are expressions with variables in the numerator or denominator. You'll learn to evaluate them for given values and simplify them by reducing, while identifying values that would make the fraction undefined.

What’s next

Next, you'll tackle interactive examples to practice evaluating and reducing these fractions, followed by a series of challenge problems to test your skills.

Property

An algebraic fraction (or rational expression, as they are sometimes called) is a fraction in which both numerator and denominator are polynomials.

Examples

• A group of friends buys a gift for $C$ dollars and splits it among 4 people. Each person's share is $\frac{C}{4}$ dollars.

• A rectangular garden has an area of 100 square meters and a width of $w$ meters. Its length is $\frac{100}{w}$ meters.

Property

We can evaluate algebraic fractions just as we do any other algebraic expression. But we cannot evaluate a fraction at any values of the variable that make the denominator equal to zero.

Examples

• To evaluate $\frac{x+5}{x-3}$ for $x=7$, we calculate $\frac{7+5}{7-3} = \frac{12}{4} = 3$.

• The fraction $\frac{10}{y+4}$ is undefined for $y=-4$ because the denominator becomes $-4+4=0$.

Property

We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one.

Examples

• We can reduce $\frac{18}{30}$ by noting $\frac{18}{30} = \frac{6 \cdot 3}{6 \cdot 5}$. We divide out the common factor 6 to get $\frac{3}{5}$.

• The fraction $\frac{5x^2}{15x}$ can be written as $\frac{5 \cdot x \cdot x}{3 \cdot 5 \cdot x}$. Canceling the common factors of $5$ and $x$ gives $\frac{x}{3}$.

Property

To Reduce an Algebraic Fraction.

1. Factor numerator and denominator completely.

2. Divide numerator and denominator by any common factors.

Property

We can cancel common factors (expressions that are multiplied together), but not common terms (expressions that are added or subtracted).

Examples

• Correct: $\frac{5(x+2)}{5} = x+2$ because 5 is a factor. Incorrect: $\frac{5x+2}{5} \neq x+2$ because 5 is not a factor of the entire numerator.

• In the fraction $\frac{x+8}{y+8}$, you cannot cancel the 8s. They are terms being added, not factors being multiplied.

Property

The opposite of $a - b$ is

Any number (except zero) divided by its opposite is $-1$. For example,

Examples

• The opposite of the binomial $5-x$ is $-(5-x) = -5+x$, which is the same as $x-5$.

• To reduce the fraction $\frac{x-10}{10-x}$, recognize that the numerator is the opposite of the denominator. The fraction simplifies to $-1$.