Lesson 2: Algebraic Fractions

New Concept

An algebraic fraction, or rational expression, is a fraction with polynomials. We'll learn to simplify them by factoring and canceling common factors—a key skill for working with rational equations and functions.

What’s next

Next, you'll tackle interactive examples and practice cards that show you exactly how to factor and reduce these fractions step-by-step.

Property

An algebraic fraction (or rational expression, as they are sometimes called,) is a fraction in which both numerator and denominator are polynomials. When working with fractions, we must exclude any values of the variable that make the denominators equal to zero.

Examples

• The algebraic fraction $\frac{x+4}{x-3}$ is undefined for $x=3$, because the denominator becomes $3-3=0$.
• For the expression $\frac{y^2+2}{y+5}$, the value $y=-5$ must be excluded because it makes the denominator zero.
• To evaluate $\frac{a-1}{2a+4}$ for $a=2$, we substitute to get $\frac{2-1}{2(2)+4} = \frac{1}{8}$.

Explanation

Think of these as fractions with variables. The most important rule is that the denominator can never equal zero, because division by zero is undefined. Always check for values that would break this rule before you start solving.

Property

Fundamental Principle of Fractions: 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.

To reduce an algebraic fraction:

1. Factor the numerator and the denominator.
2. Divide the numerator and denominator by any common factors.

Caution: We can cancel common factors, but we cannot cancel common terms.

Examples

• To reduce $\frac{12x^5y^2}{8x^3y^3}$, we find the common factor $4x^3y^2$. Factoring gives $\frac{3x^2 \cdot 4x^3y^2}{2y \cdot 4x^3y^2}$, which simplifies to $\frac{3x^2}{2y}$.
• The fraction $\frac{x+4}{x+8}$ cannot be reduced. The $x$ is a term, not a factor, so it cannot be canceled.
• To reduce $\frac{7x+14}{21}$, first factor the numerator and denominator: $\frac{7(x+2)}{7(3)}$. Canceling the common factor of 7 leaves $\frac{x+2}{3}$.

Explanation

To simplify an algebraic fraction, you must first factor the top and bottom completely. Then, you can cancel out identical factors. Remember, you can only cancel parts that are multiplied, not parts that are added or subtracted.

Property

The opposite of an expression can be found by multiplying it by $-1$. Thus, the opposite of $a - b$ is $-(a - b) = -a + b = b - a$. When a binomial is divided by its opposite, the result is $-1$.

Examples

• The fraction $\frac{x-5}{5-x}$ simplifies to $-1$ because the numerator is the opposite of the denominator.
• Reduce $\frac{4z-8}{6-3z}$. First, factor: $\frac{4(z-2)}{3(2-z)}$. The factors $(z-2)$ and $(2-z)$ are opposites, so they cancel to $-1$. The result is $\frac{4(-1)}{3} = -\frac{4}{3}$.
• The expression $\frac{x+9}{9-x}$ cannot be simplified this way. Since addition is commutative ($x+9=9+x$), the numerator and denominator are not opposites.

Explanation

When you see a subtraction in the numerator that's reversed in the denominator, like $(x-5)$ and $(5-x)$, they are opposites. When you cancel them, they don't just vanish; they simplify to $-1$.

Property

A rational function is a function defined by an algebraic fraction. That is, it has the form

where $P(x)$ and $Q(x)$ are polynomials. A rational function is undefined at any $x$-values where $Q(x) = 0$. A vertical asymptote is a vertical line on the graph that occurs where a rational function is undefined.

Examples

• The function $f(x) = \frac{20}{x-5}$ is a rational function. It is undefined when $x=5$, so its graph has a vertical asymptote at the line $x=5$.
• For the function $g(x) = \frac{x+1}{x^2-9}$, the denominator is zero when $x=3$ or $x=-3$. The graph has two vertical asymptotes: $x=3$ and $x=-3$.
• The function for average cost $C(n) = \frac{200+8n}{n}$ is rational. It is undefined for $n=0$, which means you cannot produce zero items and calculate a meaningful average cost.

Explanation

A rational function is just a fraction made of polynomials. Its graph has a special feature called a vertical asymptote—a vertical line the graph gets very close to but never crosses. This line appears wherever the denominator is zero.