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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