Cogito
AP Precalculus · Chapter 2 · Lesson 1
Zeros and Multiplicity
Whether the curve crosses the axis or bounces off it.
11 problems · about 22 minutes · AP Precalculus 1.5
What this lesson teaches
The student uses the multiplicity of a zero to predict whether the graph crosses or touches the x-axis.
- Each factor gives a zero, and its exponent is that zero’s multiplicity.
- Odd multiplicity crosses the axis; even multiplicity touches and turns back.
- The multiplicities add to the degree.
Warm Up
Straightforward practice. Get the method working first.
4 problems(x − 7)⁵. Does the graph cross at 7? 1 for yes, 0 for no.
Answer 1
Why It crosses.
(x − 1)(x + 4). How many distinct zeros?
Answer 2
Why One per distinct factor.
(x + 3)². Multiplicity of the zero at −3?
Answer 2
Why The exponent.
Sort each multiplicity by what the graph does at that zero.
Answer Crosses the axis: Multiplicity 1, Multiplicity 3 · Touches and turns back: Multiplicity 2, Multiplicity 4
Why Odd crosses, even touches.
Build It Up
The same ideas with more to keep track of.
3 problems(x − 2)²(x + 5)³. What is the degree?
Answer 5
Why Add the multiplicities.
A zero at x = 0 with multiplicity 1. Does the graph cross? 1 for yes, 0 for no.
Answer 1
Why Odd multiplicity.
(x − 4)⁶. Does the graph cross at 4? 1 for yes, 0 for no.
Answer 0
Why Even multiplicity.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSelect every zero of f(x) = x(x − 3)(x + 2).
Answer 0; 3; −2
Why Set each factor to zero.
Degree 4 with zeros at 1 and 2, both of equal multiplicity. What is each multiplicity?
Answer 2
Why They must add to 4.
The Multiplicity: In (x − 5)³(x + 2), what is the multiplicity of the zero at x = 5?
Answer 3
Why 3.
Cross or Touch: At a zero of multiplicity 4, does the graph cross the axis? 1 for yes, 0 for no.
Answer 0
Why It touches and turns back.