PreCalculus
Polynomials
Unit 3 Packet
Name___________________ Hr___
Polynomial Behavior
End behavior of a polynomial is determined by:
Sketch:
P has even degree and the leading coefficient is positive-
P has even degree and the leading coefficient is negative-
P has odd degree and the leading coefficient is positive-
P has odd degree and the leading coefficient is negative-
EXAMPLE 1: Determine the end behavior of the following functions:
𝑓(𝑥) = −2𝑥 4 + 5𝑥 3 + 4𝑥 − 7
𝑓(𝑥) = 3𝑥 5 − 6𝑥 2 + 15
𝑓(𝑥) = −4𝑥 21 − 13𝑥 5 + 34𝑥 4 + 16𝑥
EXAMPLE 2: Use zeros to graph the following polynomial functions:
1. 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 2)(𝑥 − 4)
2. 𝑓(𝑥) = −(𝑥 + 1)2 (𝑥 − 2)
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PreCalculus
Polynomials
3. 𝑃(𝑥) = 𝑥 3 − 2𝑥 2 − 3𝑥
Unit 3 Packet
4. 𝑃(𝑥) = −2𝑥 4 − 𝑥 3 + 3𝑥 2
MULTIPLICITY: the number of times a number is a ______________
Local extrema of polynomials:
EXAMPLE 3: Determine the number of local extrema by using a graphing calculator.
a) 𝑃(𝑥) = 𝑥 4 + 𝑥 3 − 16𝑥 2 − 4𝑥 + 48
b) 𝑃(𝑥) = 7𝑥 4 + 3𝑥 2 − 10𝑥
Dividing Polynomials
Label the parts of the Division Algorithm:
𝑃(𝑥) = 𝐷(𝑥) ∙ 𝑄(𝑥) + 𝑅(𝑥)
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PreCalculus
Polynomials
Unit 3 Packet
Use long division to divide: 4 12353
EXAMPLE 1: Divide the following polynomials by using long division:
a) 6𝑥 2 − 26𝑥 + 12 by 𝑥 − 4
b) 8𝑥 4 + 6𝑥 2 − 3𝑥 + 1 by 2𝑥 2 − 𝑥 + 2
Synthetic Division: Quick method to divide polynomials using only __________________

When can it be used? When divisor is of the form ____________
EXAMPLE 2: Use synthetic division to divide:
a) 2𝑥 3 − 7𝑥 2 + 5 by 𝑥 − 3
b) 3𝑥 5 + 5𝑥 4 − 4𝑥 3 + 7𝑥 + 3 by 𝑥 + 2
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PreCalculus
Polynomials
Unit 3 Packet
REMAINDER THEOREM: If the polynomial 𝑃(𝑥) is divided by _________, then the
remainder is the value ______.
EXAMPLE 3: Let 𝑃(𝑥) = 𝑥 3 − 2𝑥 2 + 6. Use the remainder theorem to find 𝑃(3).
EXAMPLE 4: Find a fifth degree polynomial with zeros at -3, 0, 4, and 5 (double root).
Real Zeros of Polynomials
Rational Zeros Theorem: Polynomials with integer coefficients have zeros of the form:
EXAMPLE 1: Find the rational zeros of 𝑃(𝑥) = 2𝑥 3 + 𝑥 2 − 13𝑥 + 6.
EXAMPLE 2: Factor the polynomial 𝑃(𝑥) = 𝑥 3 − 3𝑥 + 2.
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PreCalculus
Polynomials
Unit 3 Packet
EXAMPLE 3: Find the zeros of the polynomial and graph.
𝑃(𝑥) = 𝑥 4 − 5𝑥 3 − 5𝑥 2 + 23𝑥 + 10
EXAMPLE 4: Find the zeros of the polynomial and graph.
𝑃(𝑥) = 2𝑥 3 − 7𝑥 2 + 4𝑥 + 4
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PreCalculus
Polynomials
Unit 3 Packet
Complex Zeros
EXAMPLE 1: Find all the zeros of 𝑓(𝑥) and find the complete factorization.
a) 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 𝑥 − 3
b) 𝑓(𝑥) = 𝑥 3 − 2𝑥 + 4
EXAMPLE 2: Let 𝑓(𝑥) = 3𝑥 5 + 24𝑥 3 + 48𝑥
a) Find the complete factorization.
b) Find all 5 zeros and find the
multiplicity of each.
CONJUGATE ZEROS THEOREM: If the polynomial P had real coefficients, and if the
complex number 𝑎 + 𝑏𝑖 is a zero of P, then its ______________________ (𝑎 − 𝑏𝑖) is also
a zero of P.
EXAMPLE 3: Find a polynomial of degree 4 with zeros -2 and 0, where -2 has multiplicity 3.
EXAMPLE 4: Find a polynomial of degree 4, with zeros 𝑖, 2, and −2.
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PreCalculus
Polynomials
Unit 3 Packet
Irreducible: a quadratic polynomial with _________________________
EXAMPLE 5: Let 𝑓(𝑥) = 𝑥 4 + 2𝑥 2 − 8.
a) Factor 𝑓(𝑥) into linear and irreducible quadratic factors with REAL coefficients.
b) Factor 𝑓(𝑥) COMPLETELY into linear factors with complex coefficients.
Applications of Polynomials
EXAMPLE 1: The rabbit population on a small island is observed to be given by the
function 𝑃(𝑥) = 120𝑡 − 0.4𝑡 4 + 1000 where 𝑡 is the time (in months) since the observations
of the island began.
a) What is the maximum rabbit population on the island and when did this occur?
b) When does the rabbit population disappear from the island?
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PreCalculus
Polynomials
Unit 3 Packet
EXAMPLE 2: Snow began falling at noon on Sunday. The amount of snow on the ground at a
certain location at time 𝑡 is given by the function:
𝐴(𝑡) = 11.6𝑡 − 12.41𝑡 2 + 6.2𝑡 3 − 1.58𝑡 4 + 0.2𝑡 5 − 0.01𝑡 6
where 𝑡 is measured in days from the start of the snowfall and ℎ(𝑡) is the depth of
snow in inches.
a) What happened shortly after noon on Tuesday?
b) Was there ever more than 5 inches of snow on the ground? If so, on what
day(s)?
c) On what day and at what time (to the nearest hour) did the snow disappear
completely?
PRACTICE:
1. Let 𝑓(𝑥) = (2𝑥 − 1)(𝑥 + 3)(𝑥 − 5)
a) Find all real zeros and state the
multiplicity of each:
b) Find the y-intercept:
c) State the end behavior:
d) Estimate the local extrema:
e) Graph the function.
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PreCalculus
Polynomials
2. Let 𝑓(𝑥) = (𝑥 − 2)2 (𝑥 + 3)
Unit 3 Packet
a) Find all real zeros and state the
multiplicity of each:
b) Find the y-intercept:
c) State the end behavior:
d) Estimate the local extrema:
e) Graph the function.
3. Let 𝑓(𝑥) = 6𝑥 3 + 3𝑥 2 + 1
Use a graphing calculator to:
a) State the intervals where 𝑓 is increasing.
b) Find all local extrema.
#4 and 5: Divide using long or synthetic division:
4.
x3  2 x 2  x  3
x2
Answer:
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PreCalculus
Polynomials
Divide using long or synthetic division:
5.
Unit 3 Packet
Answer:
6 x  x  12 x  5
3x  4
3
2
6. Let 𝑓(𝑥) = 𝑥 3 − 3𝑥 2 + 3𝑥 − 1
a) Is 𝑥 − 1 a factor of 𝑓(𝑥)? Explain.
b) Evaluate 𝑓(2) using the remainder
theorem.
7. Let 𝑓(𝑥) = 𝑥 3 − 𝑥 2 − 8𝑥 + 12
a) Find all rational zeros:
b) Find all irrational zeros:
c) Sketch a graph of 𝑓.
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PreCalculus
8. Let 𝑓(𝑥) = 𝑥 4 − 6𝑥 3 + 4𝑥 2 + 15𝑥 + 4
Polynomials
Unit 3 Packet
a) Find all rational zeros:
b) Find all irrational zeros:
c) Sketch a graph of 𝑓.
9. Let 𝑃(𝑥) = 𝑥 5 + 9𝑥 3
a) State the factored form of 𝑃(𝑥).
b) Find all zeros (real and complex) and
state their multiplicity.
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PreCalculus
Polynomials
10. Let 𝑃(𝑥) = 𝑥 5 + 𝑥 3 + 8𝑥 2 + 8
Unit 3 Packet
a) State the factored form of 𝑃(𝑥).
(HINT: factor by grouping.)
b) Find all zeros (real and complex) and
state their multiplicity.
11. Between 2000 and 2010, the actual and
projected amount spent on cable television
per household per year in the US can be
modeled by:
𝐴(𝑡) = −0.213𝑡 3 + 3.96𝑡 2 + 10.2𝑡 + 366
where 𝐴 is the amount spent and 𝑡 is the
number of years since 2000.
a) Find the maximum amount that the
average household spent on cable
television. In which year did this occur?
b) Find the y-intercept. Explain what it
means in the context of the problem.
c) During which year was $455 spent per
household on TV?
12. The growth of a red oak tree can be
approximated by the function,
𝐺(𝑡) = −0.003𝑡 3 + 0.137𝑡 2 + 0.458𝑡 − 0.839,
where G is the height of the tree (in feet)
and t (2 ≤ 𝑡 ≤ 34) is its age (in years).
GRAPH:
a) Graph the function in the box to the left
and estimate the age of the tree when it
is growing most rapidly.
b) This point is called the point of
diminishing return. Why do you think it
is called this?
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PreCalculus
Polynomials
13. A rectangular package sent by a delivery
service can have a maximum combined
length and girth (perimeter of a cross
section) of 120 inches. (Assume the base
is square.)
Unit 3 Packet
a) Show that the volume of the package is
𝑉(𝑥) = 4𝑥 2 (30 − 𝑥).
b) Find the dimensions of the package that
yield the maximum volume.
c) Find a value of x such that V= 13,500.
Explain the error in this finding.
14. Graph the function 𝑓(𝑥) = 10𝑥 4 + 19.5𝑥 3 − 121𝑥 2 + 143𝑥 − 51.5 on your graphing
calculator.

Visually estimate the zero(s): 𝑥 = ______________

Use this zero to perform synthetic division to reduce the power of the polynomial.

Was your estimate accurate? Explain.

State your new polynomial:

Graph this function on your calculator. Are there any additional zeros? If so,
what are they? 𝑥 = _______________

Now graph the original function 𝑓(𝑥) in your calculator again using the window:
x: [0.5, 1.5] y:[-0.1, 0.5]. What do you notice?

Explain why you cannot always trust your calculator to determine zeros.
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PreCalculus
Polynomials
Unit 3 Packet
15. Explain how the Remainder Theorem can be used to determine if 𝑥 − 1 is a factor of
𝑥 3 − 2𝑥 2 − 11𝑥 + 12.
16. Go to the Polynomial Practice link online and solve the even polynomial equations
(showing your work).
2.
4.
6.
EXPANSION 1: Is (𝑥 − 1) a factor of 𝑥 567 − 3𝑥 400 + 𝑥 9 + 2? Explain.
EXPANSION 2: Find 𝑐 so that 4𝑥 + 3 is a factor of 20𝑥 3 + 23𝑥 2 − 10𝑥 + 𝑐.
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