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

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

Q: Given $f(x) = -x^3 - 2x^2$, what is the end behavior of the graph?

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

Q: Which of the following is NOT a possible end behavior for a quartic polynomial with a positive leading coefficient?

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

Question 1

Given the polynomial $g(x) = 6x^4 - x^2 + 7$, determine the leading term, the leading coefficient, and the degree of the polynomial. How would you classify this polynomial?

Accepted Answer

Leading term: $6x^4$, Leading coefficient: $6$, Degree: 4, Classification: quartic

Question 2

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

Accepted Answer

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

Question 3

Use substitution to determine if $x = 2$ is a zero of $f(x) = x^4 - 7x^3 + 6x^2 - 3x + 22$.

Accepted Answer

No, $f(2) 
eq 0$

Question 4

Find the zeros and their multiplicities for $f(x) = x^6 (x-4)^2 (x+3)$.

Accepted Answer

Zeros: $0$ (multiplicity 6), $4$ (multiplicity 2), $-3$ (multiplicity 1)

Question 5

For $f(x) = x^4 - 6x^3 + 4x - 3$, what is the maximum number of real zeros, $x$-intercepts, and turning points?

Accepted Answer

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

Question 6

Given $f(x) = -x^3 - 2x^2$, what is the end behavior of the graph?

Accepted Answer

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

Question 7

Use synthetic division to divide $3x^4 + 10x^3 + 4x - 4$ by $x + 3$. What is the remainder?

Accepted Answer

$-43$

Question 8

Given the rational function $f(x) = \frac{x+1}{x}$, what is the domain?

Accepted Answer

$x 
eq 0$

Question 9

For $f(x) = \frac{x-10}{x^2-4x-5}$, what are the vertical asymptotes?

Accepted Answer

$x = 5$ and $x = -1$

Question 10

Find all zeros, real and complex, of $p(x) = (x^2 - 4)(x^2 + 9)$.

Accepted Answer

$x = 2, -2, 3i, -3i$

Question 11

A polynomial of degree 5 with rational coefficients has zeros $-2 + i$, $1 + \sqrt{7}$, and $3$. What are all the remaining zeros?

Accepted Answer

$-2 - i$, $1 - \sqrt{7}$

Question 12

Find a degree 3 polynomial $p(x)$ with $x$-intercepts at $(1,0)$, $(-2,0)$, $(3,0)$, and $p(x) 	o -\infty$ as $x 	o \infty$. Write in factored form.

Accepted Answer

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

Question 13

Given the graph of a cubic polynomial with zeros at $x = -2, 0, 1$, which of the following is a possible equation for the polynomial?

Accepted Answer

$a(x+2)x(x-1)$,\ a 
eq 0$

Question 14

For the function $f(x) = \frac{x+1}{x}$, what is the horizontal asymptote?

Accepted Answer

$y = 1$

Question 15

Which of the following intervals is $f(x) = (x+5)(x+1)(x-1)$ positive?

Accepted Answer

$(-\infty, -5)$ and $(1, \infty)$

Question 16

Given $f(x) = x^3 + 5x^2 - x - 5$, what is the $y$-intercept?

Accepted Answer

$-5$

Question 17

If a polynomial function has degree $n$, what is the maximum number of turning points it can have?

Accepted Answer

$n-1$

Question 18

Which of the following is NOT a possible end behavior for a quartic polynomial with a positive leading coefficient?

Accepted Answer

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

Analyzing Polynomial Zeros and Intervals of Positivity/Negativity

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