Zeros and Multiplicity

Whether the curve crosses the axis or bounces off it.

11 problems · about 22 minutes · AP Precalculus 1.5

NameDate

Figure — use these to answer the problems

Set c so the parabola just touches the axis. That is a double zero.

y = 1x² − 4x + 4

b-4

Figure 1

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
(x − 1)(x + 4). How many distinct zeros?
Answer
(x + 3)². Multiplicity of the zero at −3?
Answer
Sort each multiplicity by what the graph does at that zero.
Write each item under the heading it belongs to: Multiplicity 1 · Multiplicity 2 · Multiplicity 3 · Multiplicity 4
Crosses the axis
Touches and turns back

The same ideas with more to keep track of.

3 problems

(x − 2)²(x + 5)³. What is the degree?
Answer
A zero at x = 0 with multiplicity 1. Does the graph cross? 1 for yes, 0 for no.
Answer
(x − 4)⁶. Does the graph cross at 4? 1 for yes, 0 for no.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

8
Select every zero of f(x) = x(x − 3)(x + 2).
- 0
- 3
- −2
- 2
- Tick every box that applies.
Degree 4 with zeros at 1 and 2, both of equal multiplicity. What is each multiplicity?
Answer
The Multiplicity
In (x − 5)³(x + 2), what is the multiplicity of the zero at x = 5?
Answer
Cross or Touch
At a zero of multiplicity 4, does the graph cross the axis? 1 for yes, 0 for no.
Answer