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

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

Q: Which of the following is a correct description of the end behavior of $$f(x) = x^7$ - 3x^5$ + 2$$?

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

Q: Which of the following is the standard form of a polynomial of degree $$3$$ with zeros at $$x = -2$$, $$x = 1$$, and $$x = 4$$?

$$f(x) = a(x+2)(x-1)(x-4)$$, where $$a \neq 0$$

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

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

Question 1

Which of the following is the correct definition of a polynomial function?

Accepted Answer

A function that can be written as $$f(x) = a_n x^n + a_{n-1} x$$^{n-1}$$ + \ldots + a_1$ x + a_0$, where all coefficients are real numbers and n$ is a non-negative integer.

Question 2

What is the degree of the polynomial $$f(x) = 4x^5$ - 2x^3$ + 7x - 9$$?

Accepted Answer

$$5$$

Question 3

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

Accepted Answer

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

Question 4

Given the polynomial $$f(x) = (x-2)^2$(x+1)$$, what is the multiplicity of the zero at $$x=2$$?

Accepted Answer

$$2$$

Question 5

If a polynomial has a zero at $$x=3$$ with multiplicity $$4$$, what happens to the graph at $$x=3$$?

Accepted Answer

The graph touches the $$x-$$axis at $$x=3$$ and bounces off.

Question 6

Which of the following is a correct description of the end behavior of $$f(x) = x^7$ - 3x^5$ + 2$$?

Accepted Answer

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

Question 7

A cubic function models the average miles per gallon of cars over several years. If the function is $$f(x) = -0.01x^3$ + 0.3x^2$ + 0.5x + 12$$, what is the degree of this polynomial and what does it tell you about the maximum number of turning points?

Accepted Answer

Degree is $$3$$; maximum number of turning points is $$2$$

Question 8

Which of the following is the standard form of a polynomial of degree $$3$$ with zeros at $$x = -2$$, $$x = 1$$, and $$x = 4$$?

Accepted Answer

$$f(x) = a(x+2)(x-1)(x-4)$$, where $$a 
eq 0$$

Question 9

If $$f(x) = x^4$ - 5x^2$ + 6$$, what are the $$y-$$intercept and the degree of the polynomial?

Accepted Answer

$$y-$$intercept is $$6$$; degree is $$4$$

Question 10

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

Accepted Answer

The graph touches the $$x-$$axis at $$x=0$$ and crosses at $$x=1$$

Question 11

A polynomial function $$f(x)$$ has zeros at $$x = -1 ($$multiplicity $$2$$), $$x = 2 ($$multiplicity $$3$$), and $$x = 4 ($$multiplicity $$1). $$How many $$x-$$intercepts does the graph have?

Accepted Answer

$$3$$

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

$$n-1$$

Question 14

Which of the following is NOT a property of a power function $$f(x) = x^n$$ where $$n is an $$even integer?

Accepted Answer

The graph is symmetric about the origin.

Question 15

Given $$f(x) = (x+1)^2$(x-3)$$, which statement about the behavior at $$x = -1 is $$correct?

Accepted Answer

The graph touches the $$x-$$axis at $$x = -1$$ and bounces off.

Question 16

Which of the following is the correct end behavior for $$f(x) = 3x^6$ - 2x^4$ + x$$?

Accepted Answer

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

Question 17

A real-world scenario: The profit $$P(x) (in $$thousands of dollars) for a company is modeled by $$P(x) = -2x^3$ + 12x^2$ - 18x + 8$$, where $$x is $$the number of products sold (in thousands). What is the degree of the polynomial, and what does this tell you about the possible number of local maxima and minima?

Accepted Answer

Degree is $$3$$; there can be up to $$2$$ local maxima or minima.

Question 18

Which of the following is true about the graph of $$f(x) = x^3$?

Accepted Answer

It is symmetric about the origin and passes through $$(0,0).$$

Question 19

If $$f(x) = x^2$ - 4x + 3$$, what are the zeros of the function?

Accepted Answer

$$x = 1$$ and $$x = 3$$

3.2 Polynomials and Power Functions: Properties, Graphs, and Applications

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