Q: Which of the following describes the end behavior of the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3$$?

As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty$$

Question 1

A pharmaceutical company models the profit $$P(x)$$ from selling $$x$$ units of a new drug as $$P(x) = -2x^2$ + 400x - 10,000. $$What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $$30,000 at$$ $$x = 100$$ units.

Question 2

Given the polynomial $$f(x) = x^3$ - 4x^2$ + x + 6$$, which of the following is a real zero of $$f(x)$$?

Accepted Answer

$$x = 3$$

Question 3

A rectangular box must have a volume of $$1000$$ cubic centimeters. If the length is twice the width and the height is $$5 cm$$, what is the width of the box?

Accepted Answer

$$10 cm$$

Question 4

Which of the following describes the end behavior of the polynomial $$f(x) = -2x^4$ + 5x^2$ - 3$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 5

A chemical reaction rate is modeled by $$R(t) = t^2$ - 6t + 8. At $$what time $$t$$ does the reaction rate reach its minimum?

Accepted Answer

$$t = 3$$

Question 6

Which of the following is the correct factorization of $$x^3$$ - $$6x^2 + 11x - 6$?

Accepted Answer

$$(x - 1)(x - 2)(x - 3)$$

Question 7

A polynomial $$f(x)$$ has zeros at $$x = 2 ($$multiplicity 2) and $$x = -1. $$Which of the following could be $$f(x)$$?

Accepted Answer

$$f(x) = (x - 2)^2$(x + 1)$$

Question 8

Which of the following is the correct horizontal asymptote for the rational function $$f(x) = \frac{3x^2$ + 2x - 1}{x^2$ - 4}$$?

Accepted Answer

$$y = 3$$

Question 9

A company wants to build a box with a square base and a volume of $$500$$ cubic inches. What is the formula for the surface area $$S in $$terms of the base side length $$x$$?

Accepted Answer

$$S = x^2$ + 4x \left( \frac{500}{x^2$} \right)$$

Question 10

Which of the following is the correct result of synthetic division of $$x^3$$ + $$2x^2 - 5x + 6$ by x - 2$?

Accepted Answer

$$x^2 + 4x + 3$ remainder 16$

Question 11

Which of the following is the correct domain of the function $$f(x) = \frac{x^2$ - 1}{x^2$ - 4}$$?

Accepted Answer

All real numbers except $$x = 2$$ and $$x = -2$$

Question 12

A polynomial $$f(x)$$ has a degree of $$5$$ and $$3$$ distinct real zeros. What is the maximum number of turning points $$f(x)$$ can have?

Accepted Answer

$$4$$

Question 13

Which of the following is the correct vertical asymptote for the function $$f(x) = \frac{2x + 1}{x^2$ - 9}$$?

Accepted Answer

$$x = 3$$ and $$x = -3$$

Question 14

A graph of a cubic polynomial passes through the origin and has a local maximum at $$x = -1$$ and a local minimum at $$x = 2. $$Which of the following could be its equation?

Accepted Answer

$$f(x) = x^3$ - 3x^2$ + 2x$$

Question 15

Which of the following is the correct oblique asymptote for $$f(x) = \frac{x^2$ + 2x + 1}{x - 1}$$?

Accepted Answer

$$y = x + 1$$

Question 16

A polynomial $$f(x)$$ has a zero at $$x = 4$$ with multiplicity $$3. $$What does the graph do at $$x = 4$$?

Accepted Answer

Crosses the $$x-$$axis

Question 17

Which of the following is the correct remainder when dividing $$2x^3$$ - $$5x^2 + 4x - 7$ by x - 2$ using synthetic division?

Accepted Answer

$$-15$$

Question 18

A rational function $$f(x) = \frac{x^2$ - 4}{x^2$ + 4}$$ has which of the following $$y-$$intercepts?

Accepted Answer

$$(0, 1)$$

Question 19

Which of the following is the correct set of complex zeros for $$x^2 + 4 = 0$?

Accepted Answer

$$x = 2i, x = -2i$$

Question 20

A company wants to maximize the surface area of a box with a fixed volume. Which formula should be used to relate surface area $$S$$ and volume $$V$$ for a box with length $$l$$, width $$w$$, and height $$h$$?

Accepted Answer

$$S = 2lw + 2lh + 2wh$$ and $$V = lwh$$

Study Notes: Polynomial and Rational Functions, Quadratics, and Related Concepts

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