Pre-Calculus
Review for Test: Polynomial Functions
Polynomial Professional
Write the equation of a polynomial function that has the characteristics listed below.
Answers may be expressed in either standard or factored form.
1. One real zero
1.
2. An absolute maximum
2.
3. A range of all real numbers
3.
4. A fourth degree polynomial function with one real zero and no non-real solutions
4.
Determine whether each statement is sometimes, always, or never true.
5. A polynomial equation of degree n will have a maximum of n roots.
5.
6. A polynomial function has three non-real solutions.
6.
7. A polynomial function with four real zeros is a fourth degree polynomial function.
7.
8. If f(x) is a polynomial function and lim f(x) = −∞, then the sign of the leading
8.
x→∞
coefficient of f(x) is positive.
Multiple Choice.
9. Which of the following characterizes the function f(x) = (x + 8)3 (x − 5)(x − 1)2 .
I. f(x) is a fifth degree polynomial
function
A.
B.
C.
D.
II. (−5,0) is a zero of f(x)
III. The sign of the leading
coefficient is positive.
I only
III only
I and II
I and III
10. If a polynomial function, f(x), has a RANGE of (−∞, 8], which of the following can
be concluded about f(x)?
I. f(x) has an even degree
A.
B.
C.
D.
9.
I only
II only
I and II
I, II, and III
II. The sign of the leading
coefficient is negative.
10.
III. f(x) has a minimum of
of two real zeros.
Given the information below, sketch a possible graph of the polynomial function, f(x).
11.
lim f(x) = ∞
12. Degree: 8
x→∞
f(2) = −5
f(−2) = 4
Relative maximum at x = −7 and x = −2
Single root at x = 4 and x = −3
Double root at x = −7
Triple root at x = 0
Relative minimum at x = −5 and x = 2
Lead Coefficient: Negative
Number of zeros: 3
Number of non-real solutions: 2
y
-10 -8 -6 -4 -2
2
4
6
8 10
x
Graph each polynomial function of the axes provided. Then answer the questions that follow.
13. f(x) = −(x + 7)2 (x + 1)(x − 5)3 (x − 8)
14. f(x) = x 3 − 2x 2 − 9x + 18
y
-10 -8 -6 -4 -2
y
2
4
6
8 10
-10 -8 -6 -4 -2
lim f(x) =
Degree:
Lead Coefficient: +
x
x→∞
or
−
lim f(x) =
x→−∞
2
4
6
8 10
x
15. Write a possible equation of least possible degree
for the polynomial function graphed below. Please
express the answer in factored form.
16. Write a polynomial equation of least degree with
the roots 8, 5i, and −5i. Please express the final
answer in standard form.
Use the graph of the polynomial function, f(x), below to answer each of the following.
When appropriate, use interval notation.
17.
a. Domain
b. Range
c. Least possible degree
d. Sign of the leading coefficient
e. Number of real zeros
f. Single root(s)
g. Double root(s)
h. Triple root(s)
i. Relative maxima
j. f(0) =
k. What is the value of the absolute maximum?
l. At what value of x does f(x)have a relative minimum?
m. Over what interval(s) is f(x) increasing?
n. Over what interval(s) is f(x) decreasing?
Use the information below to answer the Questions 18-23. Report all ages as natural numbers.
The average height (in inches) for boys ages 1 to 20 can be modeled by the function
𝐁(𝐱) = −𝟎. 𝟎𝟎𝟏𝐱 𝟒 + 𝟎. 𝟎𝟒𝐱 𝟑 − 𝟎. 𝟓𝟔𝐱 𝟐 + 𝟓. 𝟓𝐱 + 𝟐𝟓,
where x represents age in years.
18. What do x and B(x) represent in this model?
19. To the nearest inch, what is the average height of a
seven year old boy?
20. At what age will the average boy’s height by 4’2”?
21. At what age does the average boy reach his maximum
height?
22. To the nearest inch, what is the maximum average
height for boys ages 1-20?
23. Why is the domain of this function restricted to [1,20]?
Find all the complex (real and imaginary) roots of each polynomial equation.
All answers must be exact and expressed in simplest form.
24. x 4 − 4x 2 − 45 = 0
25. x 5 + 11x 4 + 38x 3 + 94x 2 + 168 = −192x