1.8 Rational Functions and Zeros

Zeros of a rational function

A rational function is a ratio of two polynomials, f(x) = p(x)/q(x). Its zeros occur where the numerator equals zero, provided that value doesn't also make the denominator zero. If a value makes both numerator and denominator zero, it may create a hole (removable discontinuity) instead of a zero after simplifying common factors.

Sign changes and graph behavior at zeros

At a zero from a factor with odd multiplicity, the graph crosses the x-axis, changing sign. At a zero with even multiplicity, the graph touches and turns back, not changing sign. Near a zero, the function's local behavior resembles that of a power function based on the multiplicity of the factor.

Finding zeros by factoring

To find zeros, factor the numerator completely and set each factor equal to zero, excluding any x-values that also zero out the denominator. Checking the denominator first helps identify excluded values (domain restrictions) that must be ruled out as true zeros, distinguishing them from asymptotes or holes.

Key terms

Zero of a rational function

A value where the numerator equals zero and the denominator does not equal zero at that same value.

Odd multiplicity zero

The graph crosses the x-axis and changes sign at that zero.

Even multiplicity zero

The graph touches the x-axis and turns back without changing sign.

Removable discontinuity (hole)

Occurs when a factor cancels between numerator and denominator; not a zero of the function.

How to find zeros of a rational function

Factor the numerator, set factors equal to zero, and exclude any values that also zero the denominator.

Try an AP-style question

What are the zeros of (x² − 9)/(x + 1)?

1. x = 9
2. x = 3 and x = −3
3. x = −1
4. x = 3, −3, and −1