Question 1

Which of the following best describes the polynomial $$f(x) = -9 - x$$? Identify its classification, leading term, leading coefficient, and degree.

Accepted Answer

Linear; leading term $$-x$$; leading coefficient $$-1$$; degree $$1$$

Question 2

Given $$g(x) = 259x^2$ + 4203x^3$, what is the degree of the polynomial and its leading coefficient?

Accepted Answer

Degree $$3$$; leading coefficient $$4203$$

Question 3

A professional basketball player has a vertical leap of $$37$$ inches. Using the formula $$V = 48T^2$, what is his hang time T$ (rounded to the nearest tenth of a second)?

Accepted Answer

$$T = 0.9$$ seconds

Question 4

For the polynomial $$f(x) = 3(x + 9)^2$(x - 9)^3$, what are the zeros and their multiplicities?

Accepted Answer

$$x = -9 ($$multiplicity $$2$$), $$x = 9 ($$multiplicity $$3)$$

Question 5

What is the maximum number of real zeros, x-intercepts, and turning points for $$f(x) = 12x^4$ - 11 + 0.12x^2$ - 7x$$?

Accepted Answer

Maximum $$4$$ real zeros, $$4 x-$$intercepts, $$3$$ turning points

Question 6

Using long division, what is the quotient and remainder when $$P(x) = 3x^3$ - x^2$ + 2x + 7 is $$divided by $$d(x) = x + 1$$?

Accepted Answer

$$Q(x) = 3x^2$ - 4x + 6$$, $$R(x) = 1$$

Question 7

A rocket is propelled upward from ground level at $$98 m/sec. $$Its height after $$t$$ seconds is $$h = -4.9t^2$ + 98t. $$During what interval of time will the rocket be higher than $$445.9$$ meters?

Accepted Answer

$$(7, 13)$$ seconds

Question 8

Given $$f(x) = x^3$ - 4x^2$ - 11x + 30$$, factor the polynomial and solve $$f(x) = 0.$$

Accepted Answer

$$(x - 2)(x + 3)(x - 5)$$; zeros: $$2$$, $$-3$$, $$5$$

Question 9

Find a polynomial function of degree $$3$$ with zeros $$2$$, $$i$$, and $$-i.$$

Accepted Answer

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

Question 10

Suppose a polynomial function of degree $$4$$ with rational coefficients has $$-2 + 5i$$ and $$5 - \sqrt{5} as $$zeros. What are the other zeros?

Accepted Answer

$$-2 - 5i$$, $$5 + \sqrt{5}$$

Question 11

Find the vertical asymptotes of $$h(x) = \frac{x^2$ - 100}{(x - 9)(x + 5)}.$$

Accepted Answer

$$x = 9$$, $$x = -5$$

Question 12

What is the horizontal asymptote of $$f(x) = \frac{x^2$ + 5x + 5}{8 - x^2$}$$?

Accepted Answer

$$y = -1$$

Question 13

For the rational function $$f(x) = \frac{7x - 6}{4x + 8}$$, what is the domain?

Accepted Answer

$$(-\infty, -2) \cup (-2, \infty)$$

Question 14

Using Descartes' Rule of Signs, how many possible positive and negative real zeros does $$P(x) = -8x^4$ - 8x^3$ - 2x^2$ - 7x + 5$$ have?

Accepted Answer

1 positive, 1 or 3 negative

Question 15

Solve the inequality $$14 - x^2$ \geq 0. $$What is the solution set?

Accepted Answer

$$[-\sqrt{14}, \sqrt{14}]$$

Question 16

For the function $$g(x) = \frac{x - 1}{x + 9}$$, solve $$g(x) \u003c 0. $$What is the solution interval?

Accepted Answer

$$(-9, 1)$$

Question 17

Given $$f(x) = x^3$ - 75x - 250$$, factor $$f(x)$$ into linear factors and state the zeros and their multiplicities.

Accepted Answer

$$(x + 5)^2$(x - 10)$$; zeros: $$-5 ($$multiplicity $$2$$), $$10$$

Question 18

If a polynomial function $$f(x) of $$degree $$4$$ with rational coefficients has $$-9i$$ and $$\sqrt{8} as $$zeros, what is the lowest degree polynomial with these zeros?

Accepted Answer

$$f(x) = x^4$ + 73x^2$ - 648$$

Precalculus Polynomial and Rational Functions Study Guide

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