Rational Functions and Zeros

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

A zero of $r(x) = p(x) / q(x)$ is a value of x where the function equals zero — which requires the numerator to vanish AND the denominator to be nonzero at that point. If both vanish, you have a hole (Topic 1.10), not a zero. AP Precalc tests this distinction directly, alongside the multiplicity rules carried over from polynomial zeros.

Zero of a rational function

Condition

r(c) = 0 \;\iff\; p(c) = 0 \;\text{AND}\; q(c) \neq 0

Multiplicity at a numerator zero

Odd multiplicity

\text{graph crosses the x-axis at the zero}

Even multiplicity

\text{graph touches and bounces}

Caution: Always check the denominator. A numerator root that also makes the denominator zero is a hole, not a zero — even though substituting gives the form

0/0

Type 1

Find zeros of a rational function

Factor numerator. List its roots. Discard any that also kill the denominator.

Example 1

Find all zeros of

r(x) = \dfrac{x^2 - 9}{x + 4}

. (A)

x = 3

only (B)

x = -3

only (C)

x = 3

and

x = -3

(D)

x = -4

x = 3

x = -3

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Type 2

Distinguish zero from hole

If a factor appears in both numerator and denominator, the corresponding x-value is a hole. If only in the numerator (with denominator nonzero there), it is a zero.

Example 2

For

r(x) = \dfrac{(x - 2)(x + 1)}{(x + 1)(x - 5)}

, classify

x = 2

and

x = -1

. (A) Both are zeros (B)

x = 2

is a zero;

x = -1

is a hole (C)

x = 2

is a hole;

x = -1

is a zero (D) Both are holes

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Type 3

Construct a rational with given zeros and asymptote

Place zeros in the numerator and the vertical asymptotes (VA) in the denominator. Make sure no factor is shared (or you'll create a hole instead of a zero/VA).

Example 3

Write a rational function with zeros at

x = 2

and

x = -1

, vertical asymptote at

x = 3

. (A)

\dfrac{(x - 2)(x + 1)}{x - 3}

(B)

\dfrac{(x + 2)(x - 1)}{x - 3}

(C)

\dfrac{x - 3}{(x - 2)(x + 1)}

(D)

\dfrac{(x - 2)(x - 1)}{x + 3}

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