Q: Which of the following is the correct end behavior for the polynomial $$f(x) = -3x^6$ + 8x^2$?

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 end behavior for the polynomial $$f(x) = -3x^6$ + 8x^2$?

Accepted Answer

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

Question 2

Given the polynomial $$g(x) = (x-3)(x^2$-9)$$, what are all the real zeros?

Accepted Answer

$$x = 3, 0, -3$$

Question 3

What is the multiplicity of the zero $$x=0$$ for the polynomial $$f(x) = -x^4$ + 4x^3$ - 4x^2$?

Accepted Answer

Multiplicity 2

Question 4

Which statement best describes the Fundamental Theorem of Algebra?

Accepted Answer

Every polynomial equation of degree $$n$$ has exactly $$n$$ complex roots, counting multiplicity.

Question 5

If a polynomial $$f(x)$$ has a zero at $$x=2$$ with multiplicity 3, what does the graph do at $$x=2$$?

Accepted Answer

It crosses the x-axis and flattens out at $$x=2.$$

Question 6

Use the Rational Root Test to determine which of the following could be a rational root of $$f(x) = 2x^3$ - 2x.$$

Accepted Answer

$$x = 1$$

Question 7

Which of the following polynomials has the same end behavior as $$h(x) = x^3$ - 2x^2$ + 26x$$?

Accepted Answer

$$g(x) = x^5$ - x$$

Question 8

Given $$f(x) = x^3$ + x^2$ - 5x - 2$$, which method would you use to find all zeros if factoring is not possible?

Accepted Answer

Long division

Question 9

A polynomial $$p(x)$$ has degree 4 and a negative leading coefficient. Which of the following describes its end behavior?

Accepted Answer

Both ends down

Question 10

If $$g(x) = x^2$ + 4$$, what are the zeros of $$g(x)$$?

Accepted Answer

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

Question 11

Which of the following is a correct application of the Linear Factorization Theorem for $$f(x) = x^3$ - 2x^2$ + 26x$$?

Accepted Answer

$$f(x) = (x - r_1$)(x - r_2$)(x - r_3$)$$, where $$r_1$$, $$r_2$$, $$r_3$$ are complex numbers

Question 12

A chemical reaction is modeled by $$C(t) = -2t^4$ + 8t^3$ - 6t^2$ + 4t. $$What is the end behavior of $$C(t) as$$ $$t 	o \infty$$?

Accepted Answer

$$C(t) 	o -\infty$$

Question 13

Which of the following is NOT a polynomial function?

Accepted Answer

$$h(x) = \frac{1}{x} + 2x$$

Question 14

A population model is given by $$P(x) = x^3$ - 6x^2$ - 8x + 24. $$What is the sum of all real zeros of $$P(x)$$?

Accepted Answer

$$-6$$

Question 15

If $$f(x) = x^4$ - 6x^2$ - 8x + 24$$, what is the degree and leading coefficient of $$f(x)$$?

Accepted Answer

Degree 4, leading coefficient 1

Question 16

Which of the following best describes the graph of a polynomial with an odd degree and a positive leading coefficient?

Accepted Answer

Falls left, rises right

Question 17

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

Accepted Answer

The graph touches and turns at $$x = -2$$, and crosses at $$x = 3$$

Question 18

Which of the following is a possible factor of $$f(x) = x^3$ + 3x^2$ - x - 3$$ according to the Rational Root Test?

Accepted Answer

$$x + 3$$

Question 19

If a polynomial $$f(x)$$ has degree 5 and all real coefficients, what is the maximum number of real zeros it can have?

Accepted Answer

5

Polynomial Functions and Their Graphs: Zeros, End Behavior, and Factoring

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