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

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

Question 1

Which of the following is a polynomial function? State its degree.

Accepted Answer

$$f(x) = 3x^5$ - 2x^3$ + x - 1$$

Question 2

What is the degree and leading coefficient of the polynomial $$f(x) = -4x^7$ + 2x^5$ - x + 9$$?

Accepted Answer

Degree: 7, Leading coefficient: -4

Question 3

Which of the following graphs could NOT represent a polynomial function?

Accepted Answer

A graph with a cusp or a gap

Question 4

Given the polynomial $$f(x) = (x+2)^2$(x-3)$$, what are the real zeros and their multiplicities?

Accepted Answer

Zero at $$x = -2 ($$multiplicity 2), zero at $$x = 3 ($$multiplicity 1)

Question 5

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

Accepted Answer

$$n - 1$$

Question 6

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

Accepted Answer

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

Question 7

A polynomial function $$f(x)$$ has zeros at $$x = -1 ($$multiplicity 3), $$x = 2 ($$multiplicity 2), and $$x = 4 ($$multiplicity 1). Which statement is true about the graph at these zeros?

Accepted Answer

The graph crosses the x-axis at $$x = -1$$ and $$x = 4$$, but touches at $$x = 2$$

Question 8

Which of the following is the standard form of the polynomial $$f(x) = x(x-2)(x+3)$$?

Accepted Answer

$$f(x) = x^3$ + x^2$ - 6x$$

Question 9

A real-world scenario: The height $$h (in $$meters) of a projectile at time $$t (in $$seconds) is modeled by $$h(t) = -5t^2$ + 20t + 1. $$What is the degree of this polynomial, and what does it tell you about the shape of the graph?

Accepted Answer

Degree 2; the graph is a parabola opening downward

Question 10

Given $$f(x) = x^4$ - 4x^2$, which of the following is a correct factorization and list of real zeros?

Accepted Answer

$$f(x) = (x^2$ - 4)x^2$ = (x-2)(x+2)x^2$; zeros at x = 0$$, 2, -2$$

Question 11

Which of the following statements about the graph of a polynomial function is FALSE?

Accepted Answer

The graph can have sharp corners or cusps

Question 12

If $$f(x) = 2(x-1)^3$(x+2)^2$, what is the y-intercept of the graph?

Accepted Answer

$$f(0) = 2(-1)^3$(2)^2 = 2(-1)(4) = -8$

Question 13

Which transformation is applied to the graph of $$y = x^3$ to obtain y$$ = $$-x^3 + 2$?

Accepted Answer

Reflect about the x-axis, then shift up 2 units

Question 14

A polynomial function $$f(x)$$ has the following end behavior: as $$x 	o \infty$$, $$f(x) 	o \infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty. $$Which of the following could be the leading term?

Accepted Answer

$$x^3$$

Question 15

Which of the following is NOT a property of the graph of a polynomial function?

Accepted Answer

The graph can have holes or gaps

Question 16

If $$f(x) = x^2$(x-1)^2(x+2$$)$$, what is the degree of f(x$$)$?

Accepted Answer

Degree 5

Question 17

Suppose a polynomial function has zeros at $$x = -3, 0, 2$$ and the graph crosses the x-axis at each zero. What is the smallest possible degree of the polynomial?

Accepted Answer

Degree 3

Question 18

Which of the following best describes the range of a power function $$f(x) = x^n$$ where $$n is an $$even integer?

Accepted Answer

All nonnegative real numbers

Polynomial Functions: Structure, Properties, and Graphs

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