AnswersAZ-Common Core Algebra II AThe Rational Roots Theorem

The Rational Roots Theorem Answers

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According to the rational root theorem, the numbers below are some of the potential roots of f(x) = 10x3 + 29x2 – 66x + 27. Select all that are actual roots.

A
Option
Option A
B
Option
Option B
C
Option
Option C
D
1
E
3
2
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p(−2) = [___]

Answer:

0

3

No, this polynomial has complex roots.No, this polynomial has irrational roots.No, this polynomial has irrational and complex roots.Yes.

A
No, this polynomial has complex roots.
B
No, this polynomial has irrational roots.
C
No, this polynomial has irrational and complex roots.
D
Yes.
4

–16–202

A
–16
B
–2
C
0
D
2
5

Question text not available

Answer:

-12

6

Which of the numbers below are some potential roots of p(x) = x3 + 6x2 − 7x − 60 according to the rational root theorem?

Answers:
Blank 1:3
Blank 2:4
8

0±2±4 ±9±3±6 ±12

Question illustration
A
0
B
±2
C
±4
D
±9
E
Option
Option E
F
±3
G
±6
H
±12
9

No, this polynomial has complex roots.No, this polynomial has irrational roots.No, this polynomial has irrational and complex roots.Yes.

A
No, this polynomial has complex roots.
B
No, this polynomial has irrational roots.
C
No, this polynomial has irrational and complex roots.
D
Yes.
10

–10 –7–531524

A
–10
B
–7
C
–5
D
3
E
15
F
24
11

Find all real solutions of this equation to answer the question.(6 – 2x)(3 – 2x)x = 40

A
Yes. Because is a root, you can cut squares with sides of in. to make the box
Option A
B
No. This equation has no real solutions.
Option B
C
No. The only real solution is x = 4. It is not possible to cut squares of this size.
Option C
12

Which of the number(s) below are potential roots of the function?p(x) = x4 + 22x2 – 16x – 12

A
±6
B
Option
Option B
C
±1
D
Option
Option D
E
±3
F
±8
13

Using the section of the graph shown, which of these potential roots of the function should you test first?f(x) = 2x3 – 9x2 – 6x + 40

A
–5
B
–2
C
1
D
8
14

According to the rational root theorem, the numbers below are some of the potential roots of f(x) = 10x3 + 29x2 – 66x + 27. Select all that are actual roots.

A
Option
Option A
B
Option
Option B
C
Option
Option C
D
1
E
3
15

0±2±4 ±9±3±6 ±12

A
0
B
±2
C
±4
D
±9
E
Option
Option E
F
±3
G
±6
H
±12
16

⇒ 0

Question illustration
Answer:

-16

17

No, this polynomial has complex roots.No, this polynomial has irrational roots.No, this polynomial has irrational and complex roots.✔ Yes.

Answers:
Selection 1:Yes.
Selection 2:No, this polynomial has complex roots.
Selection 3:No, this polynomial has irrational roots.
18

Question text not available

Answers:
Selection 1:Yes.
Selection 2:No, this polynomial has complex roots.
Selection 3:No, this polynomial has irrational roots.
19

–10 –7–531524

A
–10
B
–7
C
–5
D
3
E
15
F
24
20

Find all real solutions of this equation to answer the question.(6 – 2x)(3 – 2x)x = 40

A
Yes. Because is a root, you can cut squares with sides of in. to make the box
Option A
B
No. This equation has no real solutions.
Option B
C
No. The only real solution is x = 4. It is not possible to cut squares of this size.
Option C
21

Yuri thinks that is a root of the following function.q(x) = 6x3 + 19x2 – 15x – 28Explain to Yuri why cannot be a root.

Question illustration
Answer:

According to the Rational Root Theorem, potential rational roots must be in the form p/q, where p is a factor of the constant term (-28) and q is a factor of the leading coefficient (6). Since 3 is not a factor of 28 and 4 is not a factor of 6, 3/4 cannot be a rational root. Additionally, substituting 3/4 into the function for x does not result in 0, and dividing the polynomial by (x - 3/4) would result in a non-zero remainder, confirming that 3/4 is not a root.

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