Multiple Choice Items:

1. Which polynomial is written in standard form?

A 8x⁵ + 7x² − 5x + 4x³ − 2x⁴ − 1

B x⁴ + 6x³ − x² + 2x + 3

C 4 + x − 3x² − 9x³ + 5x⁴

D x² − 5x + x⁶ − 6x⁵ − 7

2. What degree is the following polynomial: 7x⁴ − 5x³ + 2x² − x − 6?

A 7^th

B 6^th

C 5^th

D 4^th

3. What is the name of a 3^rd degree polynomial?

A cubic

B linear

C quadratic

D quartic

4. If −9 is a root of a polynomial, what is the factor?

A x − 9

B x + 9

C x − 3

D x + 3

5. Which of the following is not a root of the polynomial x³ − 2x² − x + 2?

A − 1

B 1

C − 2

D 2

6. Which polynomial has the following roots: 0, − 3, and 6?

A x³ − 3x² − 18x

B x³ + 3x² − 18x

C x³ + 9x² − 18x

D x³ − 9x² − 18x

7. What is the end behavior of the polynomial 5x⁴ − 3x³ + 6x² − x + 4?

A up left, down right

B down left, up right

C down left, down right

D up left, up right

8. Which polynomial does not have “up to the left and down to the right” end behavior?

A −x⁵ + 2x⁴ − x³ + x² + 3x − 6

B −x⁴ + 3x³ + 5x² + x + 3

C −x⁷ − 6x⁵ + 2x² − 9x + 10

D −x + 2

9. If a polynomial has 2 local minima, 1 local maxima, and 4 roots, what is the leading coefficient (positive or negative) and what is the degree (even or odd)?

A positive, even

B positive, odd

C negative, even

D negative, odd

Multiple Choice Answer Key:

Short Answer Items:

10. What does the degree of a polynomial tell you about its graph?

11. How can you determine if a number is a root of a polynomial?

12. Why might a 4^th degree polynomial not have 4 unique roots?

13. List as many facts as you can about the polynomial whose graph is below.

Short-answer key and Scoring Rubric:

10. What does the degree of a polynomial tell you about its graph?

The degree of a polynomial tells you how many roots (x-intercepts) the polynomial’s graph has.

11. How can you determine if a number is a root of a polynomial?

You can determine if a number is a root of a polynomial by using synthetic division. If you get a remainder of 0, then that number is a root. Some students may also answer that it is a root if you plug the number into the polynomial and get 0 for the y value. This was not covered in this unit, but is also correct.

12. Why might a 4^th degree polynomial not have 4 unique roots?

A 4^th degree polynomial might not have 4 unique roots because of multiplicity. If the

graph “bounces off” at a root, that root is called a “double root.”

13. List as many facts as you can about the polynomial whose graph is below.

3^rd degree polynomial

• cubic

• roots at 3, −2, −2

• −2 is a “double root” because the graph bounces off.

• The leading coefficient is negative because the graph goes down to the right and up to the left.

• local minimum at (−2, 0)

• local maximum at (1.5, 4.5) (approximately)

• y-intercept at (0, 3)

Scoring Rubric for Question 13

Performance Assessment:

Bailey borrowed $630 from her parents to use towards buying a used car. It is common for her to borrow money, but she is good about paying her parents back. Bailey’s borrowing habit can be modeled by the polynomial

y = x⁴ − 23x³ + 181x² − 573x + 630

As Bailey’s financial advisor, it is your job to put together a portfolio for Bailey that explains her borrowing habit in detail. The portfolio should contain the following pages:

1. Title Page

2. Proof that Bailey has paid off her debt after 3 years, 7 years, and 10 years (i.e., that in these years her debt (y) is $0). Double check your work to make sure these are the only times her debt was paid off and that these years do not have multiplicity.

3. Explanations regarding the local minima [(3, 0) and (8.86, 72.81)] and maximum (5.39, 42.40) of the graph. What do these points say about Bailey’s borrowing?

4. A graph modeling Bailey’s borrowing. Label the axes and make sure to keep in mind x-values that make sense for her situation.

5. Your professional advice to Bailey regarding her borrowing habit.

Performance Assessment Scoring Rubric:

1. Title Page

2. Proof that Bailey has paid off her debt after 3 years, 7 years, and 10 years (i.e., that in these years her debt (y) is $0). Double check your work to make sure these are the only times her debt was paid off and that these years do not have multiplicity.

y is 0 at the roots. The equation is quartic which means that one of the roots must be a double root. This can be determined graphically or using synthetic division. Using 3, 7, and 10 as roots gives (x + 3)(x + 3)(x + 7)(x + 10). Multiplying this out gives the starting equation

3. Explanations regarding the local minima [(3, 0) and (8.86, –72.81)] and maximum (5.39, 42.40) of the graph. What do these points say about Bailey’s borrowing?

Local minima are when Bailey owes the least amount of money and starts borrowing again. Local maxima are when Bailey owes the most and starts paying her parents back instead of borrowing.

4. A graph modeling Bailey’s borrowing. Label the axes and make sure to keep in mind x-values that make sense for her situation.

Your professional advice to Bailey regarding her borrowing habit.

Answers will vary.