Q: Which of the following best describes the end behavior of $$P(x) = 2x^6$ + 5x^2$ - 3x + 3$$?

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

Question 1

Identify the vertex of the parabola $$f(x) = -6x^2$ + 3.$$

Accepted Answer

$$(0, 3)$$

Question 2

Find the equation of the axis of symmetry for the parabola $$f(x) = 3x^2$ + 3.$$

Accepted Answer

$$x = 0$$

Question 3

Determine the domain and range of the function $$f(x) = 6x^2$ - 2.$$

Accepted Answer

Domain: $$(-\infty, \infty)$$; Range: $$(-\infty, -2]$$

Question 4

Which of the following best describes the end behavior of $$P(x) = 2x^6$ + 5x^2$ - 3x + 3$$?

Accepted Answer

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

Question 5

Determine whether the function $$f(x) = x^4$ - 2x^2$ - 3 is $$even, odd, or neither.

Accepted Answer

Even

Question 6

What is the leading term, leading coefficient, and degree of the polynomial $$f(x) = 7x^8$ - 10 + 0.12x - x^3$?

Accepted Answer

Leading term: $$7x^8$$; Leading coefficient: $$7$$; Degree: $$8$$

Question 7

Use synthetic division to divide $$3x^2 - 8x + 6$ by x - 2$ and find the remainder.

Accepted Answer

Remainder: $$-2$$

Question 8

Given $$f(x) = \dfrac{7x - 6}{8x + 16}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x 
eq -2$$

Question 9

Find the vertical asymptote(s) of $$f(x) = \dfrac{x + 9}{x^2$ - 9}.$$

Accepted Answer

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

Question 10

Find the horizontal asymptote of $$f(x) = \dfrac{3x^2$ - 4x - 6}{5x^2$ - 5x + 5}.$$

Accepted Answer

$$y = \dfrac{3}{5}$$

Question 11

Solve the inequality $$\dfrac{x - 3}{x + 2} \u003c 0$$ and select the correct solution set.

Accepted Answer

$$(-2, 3)$$

Question 12

Given $$P(x) = 2x^3$ + 5x^2$ + 12x - 8$$, which of the following is a possible rational zero?

Accepted Answer

$$\pm1, \pm2, \pm4, \pm8, \pm\frac{1}{2}, \pm\frac{2}{3}, \pm\frac{4}{3}, \pm\frac{8}{3}$$

Question 13

Use Descartes' Rule of Signs to determine the possible number of positive real zeros for $$f(x) = 8x^3$ - 6x^2$ + 8x + 8.$$

Accepted Answer

1 or 3

Question 14

Find all rational zeros and factor $$f(x) = x^3$ + 2x^2$ - 16x - 32.$$

Accepted Answer

Rational zeros: $$-2, -4, 4$$; Factored form: $$(x + 2)(x + 4)(x - 4)$$

Question 15

Which of the following equations has a graph that is symmetric with respect to the origin only?

Accepted Answer

$$y = -3x^3$ + 7x$$

Question 16

Find the slant (oblique) asymptote of $$f(x) = \dfrac{x^2$ + 4x + 3}{x - 9}.$$

Accepted Answer

$$y = x - 5$$

Question 17

A chemical reaction is modeled by $$C(t) = -2t^2$ + 8t + 3$$, where $$C is $$the concentration at time $$t. At $$what time does the maximum concentration occur?

Accepted Answer

$$t = 2$$

Question 18

A projectile is launched and its height in meters after $$t$$ seconds is given by $$h(t) = -5t^2$ + 20t + 1. $$What is the maximum height reached by the projectile?

Accepted Answer

$$21$$ meters

Question 19

Given the polynomial $$f(x) = (x - 2)^2$(x + 6)^3$, what is the multiplicity of the zero at x = -6$?

Accepted Answer

3

Polynomial and Rational Functions: Key Concepts and Problem-Solving Strategies

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