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Polynomials - Rational Roots and Types of Roots

Total questions: 25

Worksheet time: 3hrs 43mins

Name
Class
Date
1.
Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = x3 + 2x2 - 6x + 8
a)
±1, ±8
b)
±1, ±2, ±4, ±8
c)
±1, ±2, ±4
d)
1, 2, 4, 8
2.
Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = 3x3 + 2x2 - 6x + 7
a)
±1, ±7, ±1/3, ±7/3
b)
±1, 7
c)
±1, ±3, ±7, ±7/3
d)
±1, ±3, ±7, ±1/3, ±7/3
3.
Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = 2x3 + 5x2 - 9x + 5
a)
±1, ±5, ±1/2, ±5/2
b)
±1, ±2, ±5, ±1/2, ±5/2
c)
±1, ±5
d)
±1, ±2, ±5, ±1/2, ±5/2, ±2/5
4.
Solve by using Rational Root Theorem: 
2x3 - 11x2 + 12x + 9 = 0
a)
-1, 1, 3
b)
-1, 3/2, 3 
c)
-1/2, 3, 3
d)
-1/2, 1, 3
5.
Solve by using Rational Root Theorem: 
x3 - 7x - 6 = 0
a)
1, 2, 3
b)
-1, -2, - 3
c)
-1, -2, 3
d)
1,-2,-3
6.
What should be the order of the polynomial coefficients if
(3x - 4x3 + 6x4 + 1) / (x + 3)
a)
3   0   -4   6   1
b)
6   -4   3   1   0
c)
6   -4   0   3   1
d)
-6   4   0   -3   -1
7.
If you were dividing x6 + 4x3 + 2, how many 0's would you need when setting up the top row of your synthetic division?
a)
0
b)
2
c)
4
d)
6
8.
            Is (x-2) a factor of             f(x)= x3-8x2+14x-4?
a)
Yes, (x-2) is a factor. There is a remainder.
b)
No, (x-2) is  not a factor. The remainder is zero.
c)
Yes, (x-2) is a factor. The remainder is zero.
d)
No, (x-2) is  not a factor. There is a remainder. 
9.
According to the Rational Root Theorem, what are the all possible rational roots? 
2x3 - 11x2 + 12x + 9 = 0
a)
±1,±2
b)
±1,±3,±9
c)
±1,±3±9,±1/2,±3/2,±9/2
d)
±1,±2,±1/3,±2/3,±1/9,±2/9
10.
Solve by using Rational Root Theorem: 
x3 + 3x2 - 6x - 8 = 0
a)
 -4, -1, 2
b)
-4, -1, ±√2
c)
1, 2, 4
d)
4,-1,√2
11.

How many positive solutions does the following polynomial have?

f(x) =3x5 - x4 + 3x2 - 6x +11

a)

4, 2, or 0

b)

4

c)

3 or 1

d)

3

12.

How many possible negatives solutions does the following polynomial have?

f(x) = x4 - 3x3 - 17x2 + 39x -20

a)

3 or 1

b)

3

c)

1

d)

2 or 0

13.

Use Descartes Rule of Signs to determine the possible number of positive and negative roots.

a)

1 positive, 2 or 0 negative

b)

1 positive, 1 negative

c)

0 positive, 1 negative

d)

1 positive, 0 negative

14.

Using Descartes' Rule of Signs, how many possible NEGATIVE roots are there for the polynomial f(x) = x4+x3+x2+x+5. Select all that apply.

a)

0

b)

1

c)

2

d)

3

e)

4

15.
A Polynomial of Degree 6 CAN have the following number of imaginary roots:
a)
1
b)
4
c)
3
d)
8
16.

How many roots does the following function have?

f(x)= x5 - 3x3 + x

a)

8

b)

9

c)

5

d)

3

17.

What is the name of the polynomial that has 4 roots?

a)

Quadratic

b)

Quartic

c)

Cubic

d)

Quintic

18.

A polynomial has solutions at x = 2, x = -1, and x = 3. Multiply the factors and choose the correct polynomial.

a)

x3 - 4x2 + x - 6

b)

x3 - 4x2 + x + 6

c)

x3 + 6x2 + x + 6

d)

x3 - 4x2 - x + 7

19.

What is the degree of this polynomial:

f(x)= x3(x + 3)2(x – 5)

a)

Degree: 6

b)

Degree: 5

c)

Degree: 3

d)

Degree: 2

20.

Name the following polynomial:

- 10x³ + 11x4 - 9x²

a)

quartic binomial

b)

cubic polynomial of four terms

c)

cubic binomial

d)

quartic trinomial

21.

Name the following polynomial:

3x4 + 12x² + 7x5 - 4x + 5

a)

quartic polynomial

b)

quintic polynomial

c)

quintic trinomial

d)

quartic binomial

22.

Which of the following is NOT a factor for the polynomial with roots -2, 3/4, and 5i

a)

(x + 5i)

b)

(x + 2)

c)

(3x - 4)

d)

(4x - 3)

23.

Given the function
 f(x)=−x5−5x4+12x3−60x2+27x +135f\left(x\right)=-x^5-5x^4+12x^3-60x^2+27x\ +135  
which of the following is NOT a possible combination of roots?

a)

3 real positive, 2 real negative, 0 imaginary

b)

0 real positive, 1 real negative, 4 imaginary

c)

1 real positive, 2 real negative, 2 imaginary

d)

1 real positive, 0 real negative, 4 imaginary

24.

Use Descartes Rule of Signs to determine the possible number of positive and negative roots.

a)

3 or 1 positive roots and 0 negative roots

b)

4, 2, or 0 positive roots and 1 negative root

c)

4, 2, or 0 positive roots and 0 negative roots

d)

2 or 0 positive roots and 1 negative roots

25.
Descartes' Rule of Signs is helpful is solving what kind of equation?
a)
rational
b)
exponential
c)
logarithmic
d)
polynomial

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