NAME: __________________________________________________
DATE: _________________
Algebra 2: Lesson 7-7 Zeros and Multiplicity
Learning Goals:
1) How to use zeros and multiplicity to sketch and write polynomial functions.
Directions: Look at the factors and roots of each equation and compare them to its graph. Use them fill
in the table below.
𝑓(𝑥) = (𝑥 + 1)(𝑥 − 2)2 (𝑥 − 1)
𝑓(𝑥) = (𝑥 + 2)2 (𝑥 − 1)2
𝑓(𝑥) = (𝑥 + 1)3 (𝑥 − 1)(𝑥 − 2)
𝑓(𝑥) = (𝑥 − 2)2 (𝑥 − 1)(𝑥 + 1)2
Function
𝑓(𝑥) = (𝑥 + 1)(𝑥 − 2)2 (𝑥 − 1)
Zeros
Crosses X-Axis at
Tangent to X-Axis at
Degree
𝑓(𝑥) = (𝑥 + 2)2 (𝑥 − 1)2
𝑓(𝑥) = (𝑥 + 1)3 (𝑥 − 1)(𝑥 − 2)
𝑓(𝑥) = (𝑥 − 2)2 (𝑥 − 1)(𝑥 + 1)2

What determines whether the graph crosses the x-axis or is tangent to the x-axis at a specific zero?
Zeros and Multiplicity of Polynomials

The real zeros of a polynomial function may be found by factoring (where possible) or by finding
where the graph touches the x-axis. The number of times a zero occurs is called its multiplicity. If a
function has a zero of odd multiplicity, the graph of the function crosses the x-axis at the x-value.
However, if a function has a zero of even multiplicity, the graph of the function only touches the xaxis at the x-value.

If (𝑥 − 𝑐)𝑘 is a factor of a polynomial function 𝑓(𝑥) where 𝑘 ≥ 1, and:
Directions: Determine the zeros and multiplicity of each zero for each function.
1. 𝑓(𝑥) = (𝑥 + 2)2 (𝑥 − 1)3
2. 𝑓(𝑥) = 𝑥 3 (𝑥 + 2)4 (𝑥 − 3)5
Directions: Find the zeros of each function and give the multiplicity of each zero. State whether the
graph crosses the x-axis or touch the x-axis and turns around at each zero.
4. 𝑓(𝑥) = 𝑥 3 + 5𝑥 2 − 9𝑥 − 45
3. f (x)  4x(x 1)(x  2)2

Finding a Polynomial Function Given Its Zeros and Multiplicities
Directions: Write an equation of a polynomial function which satisfies the given conditions. What is the
degree of each polynomial?
5. zeros are 0 and 3 (multiplicity of 2)
6. zeros are 1, -6, 2/3, and 4
7. zero = 2 (multiplicity 4); opens downward
8. zeros are 0, 2 (multiplicity of 2) and -4 (multiplicity
of 3).
9.
Find two polynomial functions that have the following zeros and multiplicities. What is the degree of your polynomials?
Zero
2
−4
6
−8
Multiplicity
3
1
6
10
Modeling with Polynomial Functions
Jeannie wishes to construct a cylinder closed at both ends. The figure below shows the graph of a cubic
polynomial function used to model the volume of the cylinder as a function of the radius if the cylinder is
constructed using 150𝜋 cm2 of material. Use the graph to answer the questions below. Estimate values
to the nearest half unit on the horizontal axis and to the nearest 50 units on the vertical axis.
a) What is the domain of the volume function? Explain.
b) What is the most volume that Jeannie’s cylinder can enclose?
c) What radius yields the maximum volume?
d) The volume of a cylinder is given by the formula 𝑉 = 𝜋𝑟 2 ℎ. Calculate the height of the cylinder
that maximizes the volume.