1. No category

Sample Test Questions
(Barnett, Ziegler, and Byleen)
Chapter 4
Polynomials and Rational Functions
4-1
polynomial functions and graphs
1.
For each of the following divide f(x) by g(x) and calculate the quotient q(x) and the
remainder r(x).
2.
a.
f(x) = 3x5 + 5x4 − 3x2 − x + 3,
g(x) = x − 2
(10)
b.
f(x) = 2x5 − 3x3 + 2x2 − x + 3,
g(x) = 2x + 1
(10)
c.
f(x) = x4 + 2x3 − x2 + 5,
g(x) = x2 +x + 1
(10)
d.
f(x) = x5 − 2x4 + x3 − 5,
g(x) = x2 − 2
(10)
For which of the following can you use synthetic division? Why or why not?
a.
f(x) = 3x5 + 5x4 − 3x2 − x + 3,
g(x) = x − 2
(3)
b.
f(x) = 2x5 − 3x3 + 2x2 − x + 3,
g(x) = 2x + 1
(3)
c.
f(x) = x4 + 2x3 − x2 + 5,
g(x) = x2 +x + 1
(3)
d.
f(x) = x5 − 2x4 + x3 − 5,
g(x) = x2 − 2
(3)
When you divide a polynomial of degree n, Pn(x), by a linear factor, (x−a)
3.
a.
what is the degree of the quotient?
(2)
b.
what possible remainders might you obtain?
(2)
c.
show that the remainder is equal to Pn(a)
(5)
d.
if (x−a) happens to be a factor, then what is the remainder?
(2)
Evaluate the following polynomial functions at the indicated values of x:
4.
5.
a.
f(x) = 2x3 + 3x2 − x − 1,
x = 0, −3, and i
(5)
b.
f(x) = x4 + 2x2 + 1,
x = 2, −1, and 1−i
(5)
Use synthetic division to determine the quotient and remainder for:
a.
!
3x 3 + 2x 2 " 2x + 1
x+2
(5)
2
b.
6.
7.
x4 " x2 + x " 3
x "1
A polynomial crosses the x-axis at x
(5)
= −2, 1, and 4. What are its real factors?
! A polynomial crosses the x-axis at x = 3 and just touches it at x = −2. What are its real
factors?
8.
(5)
(5)
What are the zeros of the following function?
f(x) = 4x2(3x−2)(x+2)2
(5)
9.
A polynomial function crosses the x-axis in four places. When x is large, both positive and
negative, the value of the function is positive and large. What is the degree of the
polynomial? Explain.
(5)
4-2
finding rational zeros of polynomials
10.
Find all the zeros for the following functions:
11.
a.
f(x) = 3x5 + 15x4 + 30x3 + 24x2
(15)
b.
f(x) = 2x4 + 2x3 − 14x2 − 30x
(15)
c.
f(x) = x4 − 2x3 − 2x2 − 3x
(15)
d.
f(x) = 5x4 + 40x3 + 70x2 − 40x
(15)
Hints:
are there any common factors?
how many potential positive real zeros?
how many potential negative real zeros?
what are the possible rational zeros?
what are the upper and lower bounds on zeros?
can you factor by inspection or use a formula to find any remaining zeros?
One zero of the following polynomial is
2i. Find one other zero.
(5)
2i. Find all the zeros.
(10)
g(x) = 3x3 − 2x2 + 12x − 8
12.
One zero of the following polynomial is
g(x) = 3x3 − 2x2 + 12x − 8
13.
A function can be factored as:
f(x) = (x − 2)(x2 + 3x + 4)
find its zeros.
14.
(5)
What are the possible rational zeros of each of the following functions?
a.
f(x) = x3 + 2x2 + x − 6
(5)
3
15.
b.
f(x) = 2x4 − 6x + 12
(5)
c.
f(x) = 6x4 + 3x2 − 19x − 15
(5)
Determine the number of possible real zeros, the number of negative real zeros, and the
number of possible complex zeros of:
a.
f(x) = x7 + 4x6 − 2x5 + x4 − 2x3 − 2x2 − 3x + 5
(5)
b.
f(x) = 3x7 + 2x6 − 52x5 + 31x4 − 31x3 + 23x2 − 3x + 5
(5)
That is, make a table of possible configurations of the zeros:
pos real neg real complex
zeros
zeros
zeros
16.
total
zeros
Find the integer values for the upper and lower bounds of the real zeros for:
f(x) = 4x3 + 4x2 − 9x − 9
!
4-3
approximating real zeros of polynomials
17.
Evaluating the following function at integer points yields:
(5)
f(x) = 12x3 − 40x2 − 31x + 84
a.
! b.
x
1
2
3
4
f (x)
25
"42
"45
88
5
429
x
f (x)
"1
63
"2 "110
"3 "507
"4 "1200
What are the intervals within which real zeros may be found?
(5)
Are there
! any complex zeros? Why or why not?
(5)
4
4-4
rational functions
18.
For each of the following functions, determine any vertical and horizontal asymptotes:
a.
f (x) =
3x 2 " x
x 2 " 25
(10)
b.
f (x) =
2
x +1
(10)
c.
2x 2 " x + 3
f (x) = 2
x + x "6
(10)
d.
f (x) =
2x 2 " 3
(x " 3)(x + 2)
(10)
e.
f (x) =
3x 2 " x
16 " x 2
(10)
f.
f (x) =
3x + 2
x2 +1
(10)
!
!
!
!
!
19.
For each of the following functions, determine any vertical and horizontal asymptotes and
sketch the graph of the function.
!
a.
f (x) =
3
x "2
(15)
b.
f (x) =
2x 2 " x
x2 " 9
(15)
c.
f (x) =
5
x+2
(15)
!
!
Write out a rational function that has vertical asymptotes at x = −2 and x = 3 and has an
oblique asymptote. Note: there are many possible answers—just write out one example.
20.
!
(5)