Q: Which of the following best describes the end behavior of the polynomial $$f(x) = -2x^4$ + 3x^2$ - 5$$?

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

Question 1

Which of the following best describes the end behavior of the polynomial $$f(x) = -2x^4$ + 3x^2$ - 5$$?

Accepted Answer

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

Question 2

Given $$f(x) = x^3$ - 4x^2$ + x + 6$$, which of the following is a root of $$f(x)$$?

Accepted Answer

$$x = 3$$

Question 3

Which theorem allows you to determine if $$x - a is a $$factor of a polynomial $$f(x) by $$evaluating $$f(a)$$?

Accepted Answer

Factor Theorem

Question 4

If $$f(x) = x^3$ + 2x^2$ - 5x - 6$$, use the Rational Root Theorem to determine which of the following could be a rational root.

Accepted Answer

$$x = -3$$

Question 5

Given $$f(x) = x^4$ - x^3$ - 7x^2$ + x + 6$$, which of the following is NOT a possible rational root according to the Rational Root Theorem?

Accepted Answer

$$x = 5$$

Question 6

What is the remainder when $$f(x) = 2x^3$ - 5x^2$ + 4x - 7 is $$divided by $$x - 2$$?

Accepted Answer

$$-3$$

Question 7

Which of the following statements about the graph of $$f(x) = x^2$(x-1) is $$true?

Accepted Answer

The graph touches but does not cross the x-axis at $$x = 0$$

Question 8

If $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is a factor of $$f(x)$$?

Accepted Answer

$$x - 1$$

Question 9

A polynomial $$f(x) of $$degree 5 has at most how many real zeros?

Accepted Answer

5

Question 10

Given $$f(x) = x^3$ - 2x^2$ - x + 2$$, use synthetic division to determine the remainder when divided by $$x - 2.$$

Accepted Answer

$$0$$

Question 11

Which of the following best describes the multiplicity of the zero $$x = 2$$ for $$f(x) = (x-2)^3$(x+1)$$?

Accepted Answer

Multiplicity 3, crosses the x-axis with a flattening effect

Question 12

If $$f(x) = x^4$ - 5x^2$ + 4$$, which of the following is a correct factorization over the real numbers?

Accepted Answer

$$(x^2$$ - $$4)(x^2 - 1$$)$$

Question 13

Which of the following is the correct application of the Factor Theorem?

Accepted Answer

If $$f(a) = 0$$, then $$x - a is a $$factor of $$f(x)$$

Question 14

A real-world application: The height $$h (in $$meters) of a projectile is given by $$h(t) = -5t^2$ + 20t + 1. At $$what time $$t$$ does the projectile hit the ground?

Accepted Answer

$$t = 4$$

Question 15

Given $$f(x) = x^3$ + x^2$ - 4x - 4$$, which of the following is a correct factorization?

Accepted Answer

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

Question 16

If $$f(x) = x^3$ - 7x + 6$$, which of the following is NOT a zero of $$f(x)$$?

Accepted Answer

$$x = -1$$

Question 17

Which of the following describes the possible number of turning points for a degree 4 polynomial?

Accepted Answer

At most 3

Question 18

Given $$f(x) = x^3$ - 2x^2$ - x + 2$$, which of the following is a correct application of synthetic division to test $$x = 1 as a $$root?

Accepted Answer

The remainder is 0, so $$x = 1 is a $$root.

Question 19

If $$f(x) = x^3$ - 3x^2$ + 4x - 12$$, which of the following is a possible rational root?

Accepted Answer

$$x = 3$$

Polynomial and Rational Functions: Roots, Factorization, and Theorems

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