Zeros of Rational Functions

In AP Precalculus, zeros of rational functions identify where a function’s output equals zero.
Zeros are closely connected to:

• x-intercepts of graphs

• Factoring techniques

• Domain restrictions

• Discontinuities (holes and asymptotes)

Understanding zeros of rational functions requires careful attention to both the numerator and denominator.

Definition of a Zero

A zero of a rational function is a real number $x$ such that: f(x)=0

For a rational function:

$$\frac{f(x)}{g(x)}=0 \longrightarrow f(x)=0$$ and $$g(x)\neq0$$

This means:

• Zeros come only from the numerator

• Denominator values are never allowed

Finding Zeros of Rational Functions

Step-by-Step Procedure

1. Factor the numerator and denominator completely

2. Identify values that make the numerator zero

3. Exclude any values that also make the denominator zero

Example 1: Simple Rational Function

$$f(x)=\frac{x^2-9}{x-1}$$​

Factor numerator:

$$x^2-9=(x+3)(x-3)$$

Zeros:

$$x=\pm3$$

Check denominator:

$$x\neq1$$

Final zeros:

$$x=\pm3$$

Example 2: Common Factors (Holes)

$$f(x)=\frac{x^2-4}{x-2}$$​

Factor:

$$\frac{(x-2)(x+2)}{x-2}$$​

Simplify:

$$f(x)=x+2 , x\neq2$$

• x=2 is not a zero

• A hole occurs at x=2

Zero:

$$x=-2$$

Zeros vs. Holes

Example 3: Vertical Asymptotes Do Not Create Zeros

$$f(x)=\frac{x+1}{x-3}$$​

• Zero at x=−1

• Vertical asymptote at x=3

Denominator values are never zeros.

Zeros and Multiplicity

Odd Multiplicity

• Graph crosses the x-axis

Even Multiplicity

• Graph touches the x-axis and turns

Example:

$$f(x)=\frac{(x-2)^2(x+1)}{x-3}$$​

• Zero at x=2 (even multiplicity)

• Zero at x=−1 (odd multiplicity)

Zeros and Graph Interpretation

To sketch a rational function:

1. Find zeros (x-intercepts)

2. Identify holes

3. Locate vertical asymptotes

4. Determine end behavior

Zeros anchor the graph to the x-axis.

Applications of Zeros

Zeros of rational functions model:

• Break-even points

• Equilibrium states

• Physical thresholds

• Optimization constraints

Summary

• Zeros occur when the numerator equals zero

• Denominator values are excluded

• Canceled factors create holes, not zeros

• Multiplicity affects graph behavior

• Zeros are essential for graphing rational functions