Knowledge Notes | Carey AP Precalculus Practice Lab

A numerator zero is a function zero only when the original denominator is nonzero.

What a strong response shows

Factor the entire numerator and denominator.
A zero needs numerator zero and original denominator nonzero.
Report excluded inputs separately from intercepts.

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What You Need to Know

Key idea

A rational function is zero when its numerator is zero and the denominator is not zero.

Fast decision rule

Factor both parts, cancel only for simplification, then test every candidate in the original denominator.

Worked example

A numerator zero that is excluded by the original denominator is not an x-intercept. Fully reduce the expression: a nonzero reduced denominator there gives a hole; a remaining denominator zero gives a vertical asymptote.

Common mistake

Reporting a cancelled zero as a zero of the rational function.

Mini check

What denominator check must every candidate pass?

Learning Objectives

Key idea

1.8.A determines the zeros of rational functions.

Fast decision rule

Find numerator roots, remove cancelled roots from the zero list, and keep remaining denominator roots as restrictions.

Worked steps

$1$ . Factor numerator and denominator. $2$ . Mark cancelled factors. $3$ . Solve remaining numerator factors. $4$ . Check the original denominator. $5$ . State zeros and exclusions separately.

Common mistake

Solving only the simplified numerator without the original restriction.

Mini check

Why can a graph miss an x-intercept at a hole?

Key Concepts

Key idea

Zeros, holes, and vertical asymptotes are different features even when they share an x-value in the factored form.

Fast decision rule

A remaining numerator root is a zero only if it is allowed by the original domain. At an excluded input, a nonzero fully reduced denominator gives a hole; a remaining denominator zero gives a vertical asymptote.

Worked example

If one factor cancels and another numerator factor remains, only the remaining factor creates an x-intercept.

Common mistake

Calling every numerator root an intercept.

Mini check

Which feature comes from a factor that remains in the denominator?

Relevant Representations

Key idea

Factored form and a sign chart make rational zeros easiest to locate; the graph confirms whether a candidate is present.

Fast decision rule

Use algebra to find candidates and the graph only as a check.

Classify a candidate input
Factor status	Graph feature	Is it a zero?
numerator only	x-intercept	yes
cancelled in both	hole	no
denominator only	vertical asymptote	no
constant factor	no x-value	not applicable

Common mistake

Using a hole as evidence that the function equals zero.

Mini check

What feature should you mark before making a sign chart?

Essential Relationships or Formulas

Key idea

The original domain restriction controls whether a numerator root is an actual zero.

r(x)=\frac{p(x)}{q(x)},\qquad r(c)=0\Longleftrightarrow p(c)=0\text{ and }q(c)\ne0

(x-c)\text{ cancels}\Rightarrow x=c\text{ remains excluded}

Fast decision rule

Keep the original denominator visible until all candidates are checked.

Worked example

For a numerator root $c$ with $q\left(c\right)=0$ , the input is excluded and the graph has a hole or asymptote rather than a zero.

Common mistake

Checking $q\left(c\right)$ after cancellation instead of in the original function.

Mini check

What two equations must be true for $r\left(c\right)=0$ ?

Worked Examples

Key idea

A zero test is a short sequence of factor, solve, and restriction check.

Fast decision rule

Separate the candidate list from the final zero list.

Worked steps

$1$ . Factor $p$ and q. $2$ . Cancel common factors but record them. $3$ . Solve remaining numerator factors. $4$ . Exclude any input forbidden by q. $5$ . Label holes and asymptotes.

Worked example

A rational expression with a cancelled factor at $x=2$ and a remaining numerator factor at $x=-1$ has a hole at $2$ and an x-intercept at $-1$ .

Common mistake

Returning the cancelled value as a zero.

Mini check

Which list should contain a hole’s x-value: zeros, restrictions, or both?

Common Mistakes

Key idea

Most errors are caused by cancellation, denominator restrictions, or confusing graph features.

Fast decision rule

Write “zero,” “hole,” or “asymptote” beside every candidate before answering.

Worked example

If the numerator and denominator both vanish at $c$ , substitution alone cannot classify the point; factor first.

Common mistake

Using $\frac{0}{0}$ as the function value.

Mini check

What should you do when direct substitution gives $\frac{0}{0}$ ?

AP Exam Tips

Key idea

Rational-zero questions are designed to reward careful domain bookkeeping.

Fast decision rule

Factor, cancel with a note, solve the remaining numerator, and verify the original denominator.

Worked steps

$1$ . Factor. $2$ . Record cancelled inputs. $3$ . Solve remaining numerator factors. $4$ . Check restrictions. $5$ . Report zeros separately from holes and asymptotes.

Common mistake

Giving only x-values without naming the graph feature.

Mini check

Does your answer distinguish an intercept from a missing point?

Practice

Use the Questionbank items below to find and classify rational-function zeros.

TRY IT IN A NEW SETTING

Can you transfer the method?

Find the zeros of $r(x)=(x-1)(x+2)/((x-1)(x-4))$ .

Compare your reasoning

The original domain excludes $1$ and $4$ . The only zero is $x=-2$ , where the original denominator is nonzero. The excluded input $x=1$ is not a zero.

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

1.8 · Rational Functions and Zeros

Topic 1.8 · Rational Functions and Zeros

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?