Question 1

A pharmaceutical company models the concentration of a drug in the bloodstream over time with $$f(x) = -2x^4$ + 3x^2$ - 1. $$What is the degree of this polynomial, its end behavior, and the maximum number of turning points?

Accepted Answer

Degree: 4; End behavior: both ends down; Max turning points: 3

Question 2

A population growth model uses $$f(x) = x^5$ - 2x^3$ + x. $$What is the maximum number of zeros and turning points this polynomial can have?

Accepted Answer

Maximum zeros: 5; Maximum turning points: 4

Question 3

Consider $$f(x) = -x^3$ + 4x$$, which models the velocity of a car over time. What is the end behavior of this polynomial?

Accepted Answer

Left up, right down

Question 4

A chemical reaction rate is modeled by $$f(x) = x^3$ - x^2$ - 6x. $$What are the zeros of this polynomial?

Accepted Answer

$$x = 0, x = 3, x = -2$$

Question 5

A business models profit with $$f(x) = (x - 2)(x + 1)(x - 4). $$What are the zeros of this polynomial?

Accepted Answer

$$x = 2, x = -1, x = 4$$

Question 6

Using the Factor Theorem, is $$(x - 1) a $$factor of $$f(x) = x^3$ - 3x^2$ + x + 1$$?

Accepted Answer

Yes, because $$f(1) = 0$$

Question 7

A physics experiment uses $$f(x) = (x - 3)^2$(x + 2). $$What are the zeros and what is the behavior at each zero?

Accepted Answer

$$x = 3 ($$bounce), $$x = -2 ($$cross)

Question 8

For $$f(x) = (x + 1)^3$(x - 2)^2$, which models the temperature change in a reaction, describe the behavior at each x-intercept.

Accepted Answer

$$x = -1 ($$cross), $$x = 2 ($$bounce)

Question 9

Without graphing, analyze $$f(x) = - (x - 1)^2$ (x + 3)$$, which models the height of a projectile. What is the degree, end behavior, and behavior at each zero?

Accepted Answer

Degree: 3 (odd); End behavior: left up, right down; $$x = 1 ($$bounce), $$x = -3 ($$cross)

Question 10

For $$f(x) = (x - 2)^2$ (x + 1)(x - 4)$$, which models the stress on a bridge, answer: What is the degree, end behavior, zeros and their multiplicities, and maximum number of turning points?

Accepted Answer

Degree: 4; End behavior: both ends up; Zeros: $$x = 2 ($$mult 2, bounce), $$x = -1 ($$cross), $$x = 4 ($$cross); Max turning points: 3

Question 11

A polynomial $$f(x) = x^4$ - 5x^2$ + 6 is $$used to model the spread of a virus. What is the maximum number of x-intercepts and turning points?

Accepted Answer

Maximum x-intercepts: 4; Maximum turning points: 3

Question 12

A polynomial $$f(x) = (x - 5)^3$ models the pressure in a tank. What is the behavior at x = 5$?

Accepted Answer

The graph crosses the x-axis at $$x = 5$$

Question 13

A polynomial $$f(x) = (x + 2)^4$ models the temperature in a chemical process. What is the behavior at x = -2$?

Accepted Answer

The graph touches and bounces at $$x = -2$$

Question 14

A polynomial $$f(x) = -x^6$ + 2x^3$ - 5$$ models the cost of production. What is the end behavior of this polynomial?

Accepted Answer

Both ends down

Question 15

A polynomial $$f(x) = x^2$(x - 1)^3$ models the height of a roller coaster. What are the zeros and their multiplicities?

Accepted Answer

$$x = 0 ($$mult 2), $$x = 1 ($$mult 3)

Question 16

A polynomial $$f(x) = (x - 2)^2$(x + 3)^2$ models the voltage in a circuit. What is the behavior at each zero?

Accepted Answer

$$x = 2 ($$bounce), $$x = -3 ($$bounce)

Question 17

A polynomial $$f(x) = -3x^5$ + 2x^2$ - 7$$ models the population of a species. What is the end behavior?

Accepted Answer

Left up, right down

Question 18

A polynomial $$f(x) = (x - 1)^4$(x + 2)$$ models the profit of a company. What is the degree, and what is the behavior at each zero?

Accepted Answer

Degree: 5; $$x = 1 ($$bounce), $$x = -2 ($$cross)

Polynomial Functions: Key Concepts and Practice

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