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 $x \to -\infty$, $P(x) \to \infty$

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

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

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

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

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 $\dfrac{x - 3}{x + 2} < 0$ 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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