Rational Functions and Zeros

Welcome, students! 📘 In this lesson, you will learn how rational functions work, how to find their zeros, and why those zeros matter in AP Precalculus. Rational functions show up whenever one polynomial is divided by another polynomial, and they are a key part of the unit on Polynomial and Rational Functions. In real life, they can model rates, concentrations, and efficiency, so understanding them helps you make sense of changing quantities in the world around you 🌍.

Objectives for this lesson:

• Explain the main ideas and vocabulary behind rational functions and zeros.
• Use reasoning and procedures to find zeros of rational functions.
• Connect zeros to graphs, factors, and restrictions on the domain.
• Summarize how rational functions fit into the larger study of polynomial and rational functions.
• Support your thinking with examples that match AP Precalculus style.

What Is a Rational Function?

A rational function is a function that can be written as a quotient of two polynomials. In other words, it has the form

$$f(x)=\frac{p(x)}{q(x)}$$

where $p(x)$ and $q(x)$ are polynomials, and $q(x)\neq 0$.

This condition matters because division by zero is undefined. That means the domain of a rational function must exclude any value of $x$ that makes the denominator equal to $0$.

For example,

$$f(x)=\frac{x+2}{x-5}$$

is a rational function. The denominator is $x-5$, so $x=5$ is not allowed in the domain.

Rational functions are useful because they describe situations where one quantity is compared to another. For example, if you divide distance by time, you get speed. If the time changes, the speed may change too. Rational functions often describe that kind of relationship.

Important vocabulary

• Polynomial: an expression like $x^2-3x+4$.
• Rational function: a ratio of two polynomials.
• Domain: all input values allowed in a function.
• Zero: an input value that makes the function output $0$.
• $x$-intercept: a point where the graph crosses or touches the $x$-axis, which happens when the function value is $0$.

Finding Zeros of Rational Functions

The zero of a function is where the output is $0$. For a rational function, this means solving

$$\frac{p(x)}{q(x)}=0$$

A fraction equals $0$ only when its numerator is $0$ and its denominator is not $0$. So to find zeros of a rational function, you set the numerator equal to zero and then check that the denominator is not zero at those values.

For example, let

$$f(x)=\frac{(x-3)(x+1)}{x-5}$$

To find the zeros, set the numerator equal to zero:

$$(x-3)(x+1)=0$$

This gives $x=3$ or $x=-1$.

Now check the denominator:

$$x-5\neq 0$$

At $x=3$ and $x=-1$, the denominator is not zero, so both are valid zeros.

So the zeros are $x=3$ and $x=-1$.

That means the graph has $x$-intercepts at $(3,0)$ and $(-1,0)$.

A common mistake

A common mistake is to set the whole fraction equal to zero and try to solve by cross-multiplying. That is not the right first step for zeros. Instead, remember this rule:

$$\frac{p(x)}{q(x)}=0 \quad \text{only when} \quad p(x)=0 \text{ and } q(x)\neq 0$$

This is one of the most important ideas in the topic.

Factoring, Cancelling, and What Zeros Really Mean

Many rational functions can be factored. Factoring helps you see zeros and restrictions more clearly.

Consider

$$f(x)=\frac{(x-2)(x+4)}{(x-2)(x-1)}$$

At first glance, it looks like the factor $x-2$ appears in both the numerator and denominator. If you simplify, you get

$$f(x)=\frac{x+4}{x-1}$$

but this simplification does not mean $x=2$ is allowed. The original function still has a denominator of $0$ when $x=2$, so $x=2$ is excluded from the domain.

This creates an important idea: a value may be removed from the domain even if it cancels algebraically. That point is called a hole or removable discontinuity.

For zeros, only values that make the numerator zero and are still allowed in the domain count.

So in the original function above:

• $x=2$ is not a zero, because the function is undefined there.
• If we solve the simplified numerator $x+4=0$, then $x=-4$ is a zero.

Why this matters on graphs

A rational function can have:

• zeros where the graph crosses the $x$-axis,
• holes where the graph has an open circle,
• vertical asymptotes where the graph becomes unbounded.

These features are different, and zeros should not be confused with holes or asymptotes. 🎯

Graphing Connections: Zeros and the $x$-Axis

The zeros of a rational function are directly connected to the graph because each zero corresponds to an $x$-intercept. If $f(a)=0$, then the graph passes through $(a,0)$.

Let’s look at an example:

$$f(x)=\frac{x^2-9}{x+2}$$

Factor the numerator:

$$f(x)=\frac{(x-3)(x+3)}{x+2}$$

The zeros come from the numerator:

$$x-3=0 \quad \text{or} \quad x+3=0$$

So the zeros are $x=3$ and $x=-3$.

The denominator gives a restriction:

$$x+2\neq 0$$

so $x\neq -2$.

That means the graph has two $x$-intercepts, but it also has a vertical asymptote at $x=-2$.

Sometimes students think the denominator can create zeros. It cannot. The denominator creates restrictions and may create asymptotes, but zeros come from the numerator.

Real-world thinking

Suppose a company’s profit per unit is modeled by a rational function. If the function is zero, that means the company breaks even. Finding zeros helps identify break-even points, which is important in business planning. 💼

AP Precalculus Reasoning: Using Structure and Evidence

AP Precalculus asks you to reason about functions, not just calculate. That means you should use the structure of expressions to make conclusions.

Suppose

$$g(x)=\frac{(x-1)^2(x+5)}{(x-4)(x+5)}$$

First, look for zeros and restrictions.

• The numerator is zero when $x=1$ or $x=-5$.
• The denominator is zero when $x=4$ or $x=-5$.

So:

• $x=1$ is a zero.
• $x=-5$ is not a zero, because it makes the denominator $0$ too.
• $x=4$ is not in the domain.

After canceling the common factor $x+5$, the simplified form is

$$g(x)=\frac{(x-1)^2}{x-4}$$

but the original domain restriction $x\neq -5$ still remains. That means the graph has a hole at $x=-5$.

This example shows a key AP skill: use algebraic structure to determine graph features and explain why they happen.

A useful checklist

When analyzing rational functions, ask:

1. What values make the denominator $0$?
2. What values make the numerator $0$?
3. Do any factors cancel?
4. Which zeros are valid and which are excluded?
5. Where are the intercepts, holes, and asymptotes?

Using this checklist can help you avoid errors and communicate clearly.

Conclusion

Rational functions are ratios of polynomials, and their zeros are the $x$-values that make the function equal to $0$. To find zeros, focus on the numerator, but always check that the denominator is not $0$ at those values. Factoring helps you identify zeros, domain restrictions, holes, and asymptotes. These ideas are central to AP Precalculus because they connect algebraic expressions to graphs and real-world situations. If you understand how zeros work in rational functions, you are building a strong foundation for the broader study of polynomial and rational functions. Keep practicing, students, and remember that careful reasoning matters as much as calculation 🚀.

Study Notes

• A rational function has the form $f(x)=\frac{p(x)}{q(x)}$, where $p(x)$ and $q(x)$ are polynomials and $q(x)\neq 0$.
• The domain excludes any $x$-value that makes the denominator $0$.
• A zero of a rational function is a value of $x$ that makes the function equal to $0$.
• To find zeros, set the numerator equal to $0$ and make sure the denominator is not $0$ there.
• Zeros correspond to $x$-intercepts on the graph.
• A canceled factor can create a hole, not a zero.
• A denominator factor that remains uncanceled can create a vertical asymptote.
• Factoring is one of the best tools for analyzing rational functions.
• Always check both numerator and denominator before naming a zero.
• Rational functions are important in AP Precalculus because they connect algebraic structure, graphs, and real-world modeling.