Advanced Algebra 2
Intro to Polynomials Packet
Poly – ________________________________________
Date: _______________________ Period: ___________
A polynomial function is a ________________________ function. There are no breaks, ___________________, or gaps.
We have discussed two types of functions that are polynomials already this year. With your group discuss what the two
types might be and sketch both of them below
A polynomial function also must have smooth, rounded turns. They CANNOT have _________________________ turns.
Polynomials can be difficult to graph because they contain many turns and different sized curves. Use your graphing
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9
utility to graph 𝑓(𝑥) = 5 𝑥 5 − 2𝑥 3 + 5.
How many turns in the graph are there? ________________________________________
How many real zeros can you see? _____________________________________________
Part 1 – Graphing Polynomials
The __________________________, which is defined as the highest _____________ of a polynomial, as well as, the
leading __________________________ tell us a lot about the graph. We will discuss all possible graphs below.
ODD DEGREES
a.)
Graph 𝑓(𝑥) = 𝑥 5 − 𝑥 in your graphing utility
and sketch below.
b.)
Graph 𝑓(𝑥) = −𝑥 3 + 4𝑥 in your graphing
utility and sketch below.
EVEN DEGREES
c.)
Graph 𝑓(𝑥) = 𝑥 4 − 5𝑥 2 + 4 in your graphing
utility and sketch below.
d.)
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Graph 𝑓(𝑥) = − 4 𝑥 4 + 3𝑥 2 in your graphing
utility and sketch below.
CASE
END BEHAVIOR OF GRAPH
When degree is ODD and lead coefficient is POSITIVE
Graph _______ to the left and _______ to the right
When degree is ODD and lead coefficient is NEGATIVE
Graph _______ to the left and _______ to the right
When degree is EVEN and lead coefficient is POSITIVE
Graph _______ to the left and _______ to the right
When degree is EVEN and lead coefficient is NEGATIVE
Graph _______ to the left and _______ to the right
𝐴𝑠 𝑥 → −∞ 𝐴𝑠 𝑥 → ∞
Part 2 – Finding Zeros
We know how to find the zeros of a quadratic function (I hope)! Let’s use our prior skills to help with polynomials.
Ex 1.)
Take the polynomial l𝑓(𝑥) = 𝑥 2 − 4.
a.)
We know that there are _____ solution(s) to this polynomial. Please list them below and plot them on the
provided graph.
Solution(s): ________________
b.)
State the end behavior for the polynomial.
𝐴𝑠 𝑥 → −∞, 𝑓(𝑥) → _______
𝐴𝑠 𝑥 → ∞, 𝑓(𝑥) → _______
c.)
Pick a value for x between the solution(s) and evaluate it.
x
f(x)
d.)
Sketch your graph.
Ex. 2.) How about the polynomial 𝑓(𝑥) = −2𝑥 4 + 2𝑥 2
a.)
We know that there are _____ solution(s) to this polynomial. Please list them below and plot them on the
provided graph.
Solution(s): ________________
b.)
State the end behavior for the polynomial.
𝐴𝑠 𝑥 → −∞, 𝑓(𝑥) → _______
𝐴𝑠 𝑥 → ∞, 𝑓(𝑥) → _______
c.)
Pick a value for x between the solution(s) and evaluate it.
x
f(x)
d.)
Sketch your graph.
You try…
Ex. 3.) ℎ(𝑡) = 𝑡 3 − 4𝑡 2 + 4𝑡
a.)
We know that there are _____ solution(s) to this polynomial. Please list them below and plot them on the
provided graph.
Solution(s): ________________
b.)
State the end behavior for the polynomial.
𝐴𝑠 𝑥 → −∞, 𝑓(𝑥) → _______
𝐴𝑠 𝑥 → ∞, 𝑓(𝑥) → _______
c.)
Pick a value for x between the solution(s) and evaluate it.
t
h(t)
d.)
Sketch your graph.
Ex. 4.) 𝑔(𝑡) = 𝑡 4 − 𝑡 3 − 20𝑡 2
a.)
We know that there are _____ solution(s) to this polynomial. Please list them below and plot them on the
provide graph.
Solution(s): ________________
b.)
State the end behavior for the polynomial.
𝐴𝑠 𝑥 → −∞, 𝑓(𝑥) → _______
𝐴𝑠 𝑥 → ∞, 𝑓(𝑥) → _______
c.)
Pick a value for x between the solution(s) and evaluate it.
t
g(t)
d.)
Sketch your graph.
Sketch the following polynomials
A.)
𝑓(𝑥) = (𝑥 − 4)(𝑥 − 4)
B.)
𝑓(𝑥) = (𝑥 + 3)(𝑥 + 3)(𝑥 − 5)
C.)
𝑓(𝑥) = (𝑥 − 7)(𝑥 + 2)(𝑥 − 7)
D.)
𝑓(𝑥) = (𝑥 + 1)(𝑥 + 1)(𝑥 + 1)
Repeated Zeros of Polynomial
If the root is repeated an _____________ # of times, the graph “_______________” the x-axis at that point.
If the root is repeated an ____________ # of times, the graph “________________” off the x-axis at that point.
Part 3: Practice! Practice! Practice!
Sketch the following:
E.)
𝑓(𝑥) = (𝑥 + 3)(𝑥 − 2)3 (𝑥 − 5)2
F.)
𝑓(𝑥) = (𝑥 + 2)3
Given the following polynomials state the end behavior for each without graphing them.
G.)
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𝑓(𝑥) = 3 𝑥 3 + 5𝑥
Lead Coefficient: __________________
Degree: _________________________
The graph _____________ to the left and ____________ to the right.
H.)
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𝑔(𝑥) = 5 − 𝑥 − 3𝑥 2
Lead Coefficient: __________________
Degree: _________________________
The graph _____________ to the left and ____________ to the right.
Part 4: Relative Extrema
When we graphed quadratic equations we discussed how the vertex represented a ____________ or ____________.
Graph the following equation using your graphing utility 𝑓(𝑥) = 𝑥 3 − 5𝑥 2 + 3𝑥 + 2. Sketch the graph below.
1.)
With your group discuss what you notice about the max and
min for this polynomial function.
2.)
We call this the _____________________________
or _____________________________
You try. Using your graphing utility, what are the
relative max and/or min of 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 5.
_______________________________________
_______________________________________
You try. Using your graphing utility, what are the
relative max and/or min of 𝑔(𝑥) = −𝑥 4 − 2𝑥 3 + 2𝑥 2 + 4𝑥
_______________________________________
_______________________________________
Part 5 – Evaluating Polynomial Functions
Given polynomials 𝒑(𝒙) = 𝒙𝟑 + 𝟒𝒙𝟐 − 𝟓𝒙 and 𝒈(𝒙) = 𝒙𝟐 + 𝟑𝒙 + 𝟒. Evaluate the following…
1.)
Find 𝑝(−4)
__________________________________________________________________________________________________
2.)
Find 𝑝(2𝑎3 )
__________________________________________________________________________________________________
3.)
Find 𝑔(𝑚 − 2)
__________________________________________________________________________________________________
4.)
Find g(𝑐 + 1) − 2𝑝(𝑐)