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Algebra II – Chapter 6 Day #2
Topic: Linear Factors & Zeros
Today, we will continue our study of polynomial functions.
Standards/Goals:
 F.IF.7.c: I can use the graph of a polynomial function to find the zero’s.
 I can understand the relationship between zeros, roots, and x-intercepts.
 I can write polynomials in factored form.
 I can find the multiplicity of the graph of a polynomial.
We want to first take a look back at the idea of finding a ‘degree.’
What is the degree of the polynomial function that generates the data shown in the table shown?
X -3 -2 -1 0 1 2 3
Y -1 -7 -3 5 11 9 -7
We also want to revisit the idea of factoring in order to write a polynomial in ‘factored form.’
Example: What is the factored form of 𝑥 3 − 2𝑥 2 − 15𝑥?
Now, try this one: What is the factored form of 𝑥 3 + 𝑥 2 − 12𝑥?
We also want to tackle the issue of what ‘zero’s are.
MEMORIZE THESE FACTS ABOUT ZERO’S:
a. The words: Zeros, Roots, and Solutions are all synonyms. (They all mean essentially the same
thing).
b. The three words just mentioned refer to the numbers that make a polynomial equal to 0 when
those numbers are substituted in for x.
c. You can see the REAL zeros, by looking at the graph of a polynomial because the REAL zeros are
the x-intercepts on the graph.
d. Polynomials have a varying number of zero’s because of a couple things. The way we tell how
many zeros a polynomial has is based on the highest exponent. The largest number of the
exponents is known as the ‘DEGREE’ of the polynomial AND this tells us the number of zeros that
you are looking for. This number will include both the REAL and IMAGINARY zeros.
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EXAMPLE: Consider the following polynomial. How many zeros will this polynomial have?
𝒙𝟓 + 𝒙𝟒 + 𝒙𝟑 + 𝒙𝟐 + 𝒙 + 𝟏𝟎 = 𝟎
Answer: This polynomial would have 5 zeros and some of which would be imaginary. It should be
noted that an imaginary zero is NOT an x-intercepts.
Example: Consider this following polynomial. How many zeros will this polynomial have?
𝒙𝟐 + 𝟔𝒙 − 𝒙𝟒
We also want to be able to find the zeros of a polynomial function and use those zero’s to graph
the function.
Example: What are the zeros of y = (x + 2) (x – 1) (x – 3)? Graph the function.
 Use the Zero Property to find the zeros
 Find the points for x-values between the zeros
 Determine the end behavior
 Use the zeros to sketch the graph.
We also want to be able to write a polynomial function from its zeros.
Example: What is a cubic polynomial function in standard form with zeros at 1, -1, and 4?
What is a quartic polynomial function in standard form with zeros at -2, -2, 2, and 3?
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We also want to tackle the issue of ‘multiplicity’.
Multiplicity – A way to express the fact that some zeros appear more than one time as a factor.
Example: What are the zeros of 𝑔(𝑥) = (𝑥 + 2)2 (𝑥 − 2)(𝑥 − 3)?
Consider the factored polynomial function:
f(x) = (2𝑥 − 5)2 (𝑥 + 2)
Identify the zeros of this function and state the multiplicity of each.
a. _____________ is a zero of multiplicity _____________.
b. _____________ is a zero of multiplicity _____________.
REMEMBER THIS ABOUT MULTIPLICTY (when looking at the graph):
a. If the graph touches the x-axis then changes direction that zero has an EVEN multiplicity.
b. If the graph passes through the x-axis then that zero has an ODD multiplicity.
Example:
What are the zeros of 𝑓(𝑥) = 𝑥 3 − 5𝑥 2 + 3𝑥 + 9? What are the multiplicities? How does the
graph behave at these zeros?
We finally, want to discuss the issues of relative maximum and minimums when looking at
the graph of a polynomial function.
You should think of these two things as the ‘turning points’ of the graph. (See the graph
below).
Example: Consider the graph below.
Determine the relative maximum and/or minimum.
Example: What are the relative maximum and minimum of 𝑓(𝑥) = 𝑥 3 − 9𝑥? Round answer to the
nearest tenth.
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HOMEWORK – Chapter 6 Day #2
Determine the degree of the polynomial function with the given data.
1.
X -3 -2 -1 0 1 2 3
Y -43 -10 1 2 5 22 65
2.
X -3 -2 -1 0 1 2 3
Y 65 5 -5 -1 5 25 95
Write each polynomial in factored form.
3. 𝑥 3 − 3𝑥 2 − 𝑥 + 3
4. 𝑥(𝑥 2 + 11𝑥 + 30)
5. 𝑥 3 + 9𝑥 2 + 18𝑥
6. How many zeros do each of the following polynomials have?
a. 𝑥 3 + 5𝑥 − 𝑥 7
b. 𝑥 2 + 9𝑥 − 6𝑥 5 − 6𝑥 3 + 10
What are the zeros of the following functions? Graph the function.
#7. y = x(x - 1)(x + 3)
#8. y = (x + 2)(x + 3)
9. Write a polynomial function in standard form with the given zeros:
a. x = -2, 1, 4
b. x = 3, -8, 0
c. x = 3, 0
10. Find the zeros of each function. State the multiplicity of multiple zeros.
a. y = (𝑥 − 3)2 (𝑥 + 1)
b. y = (𝑥 + 5)6 (𝑥 − 9)3 (𝑥 + 17)
c. y = 2𝑥 2 − 2𝑥
11. Using your graphing calculator, find the relative minimum and relative maximum
of the graph of each function.
a. f(x) = −3𝑥 3 + 10𝑥 2 + 6𝑥 − 3
b. f(x) = 𝑥 3 + 4𝑥 2 − 𝑥 + 1