Q: Given the polynomial function $$f(x) = 5x^5$ - 4x^3$ + x$$, determine the end behavior of the function as $$x \to \infty$$ and $$x \to -\infty.$$

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

Q: Which of the following best describes the end behavior of a polynomial with an even degree and a negative leading coefficient?

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

Q: A polynomial function $$f(x)$$ has zeros at $$x = -2 ($$multiplicity 1), $$x = 1 ($$multiplicity 2), and $$x = 3 ($$multiplicity 3). Which zero(s) will the graph cross the x-axis?

$$x = -2$$ \, \text{and} \, x = 3$

Q: Suppose $$f(x) = x^4$ - 5x^2$ + 4. $$What is the end behavior of $$f(x)$$?

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

Q: A chemical reaction's rate is modeled by $$f(x) = -2x^4$ + 3x^2$ - x + 5. $$What is the end behavior of this function?

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

Question 1

Given the polynomial function $$f(x) = 5x^5$ - 4x^3$ + x$$, determine the end behavior of the function as $$x 	o \infty$$ and $$x 	o -\infty.$$

Accepted Answer

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

Question 2

Which of the following best describes the end behavior of a polynomial with an even degree and a negative leading coefficient?

Accepted Answer

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

Question 3

Given the polynomial $$f(x) = (x-2)^3$(x+1)^2$, what are the zeros and their multiplicities?

Accepted Answer

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

Question 4

If a polynomial function has a zero at $$x=4$$ with multiplicity 2, what happens to the graph at $$x=4$$?

Accepted Answer

The graph touches the x-axis at $$x=4$$ but does not cross it.

Question 5

How many turning points can the graph of a degree 5 polynomial have at most?

Accepted Answer

$$4$$

Question 6

Given $$f(x) = x^3$ - 6x^2$ + 11x - 6$$, which of the following is NOT a zero of $$f(x)$$?

Accepted Answer

$$x = 4$$

Question 7

A polynomial function $$f(x)$$ has zeros at $$x = -2 ($$multiplicity 1), $$x = 1 ($$multiplicity 2), and $$x = 3 ($$multiplicity 3). Which zero(s) will the graph cross the x-axis?

Accepted Answer

$$x = -2$$ \, 	ext{and} \, x = 3$

Question 8

Which of the following statements about the Intermediate Value Theorem is correct?

Accepted Answer

If $$f(a)$$ and $$f(b)$$ have opposite signs, then $$f(x)$$ has at least one zero in $$[a, b].$$

Question 9

Suppose $$f(x) = x^4$ - 5x^2$ + 4. $$What is the end behavior of $$f(x)$$?

Accepted Answer

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

Question 10

A chemical reaction's rate is modeled by $$f(x) = -2x^4$ + 3x^2$ - x + 5. $$What is the end behavior of this function?

Accepted Answer

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

Question 11

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

Accepted Answer

The graph touches the x-axis at $$x = -2$$ and turns around.

Question 12

A polynomial $$f(x) is $$even if:

Accepted Answer

$$f(-x) = f(x)$$ for all x

Question 13

Which of the following polynomials is odd?

Accepted Answer

$$f(x) = x^3$ - x$$

Question 14

If $$f(x) = (x-1)^2$(x+3)$$, what is the degree of the polynomial?

Accepted Answer

$$3$$

Question 15

A medical dosage model is given by $$f(x) = -x^3$ + 6x^2$ - 9x + 4. $$What is the maximum number of turning points this function can have?

Accepted Answer

$$2$$

Question 16

Given $$f(x) = x^3$ - 2x^2$ - x + 2$$, use the Intermediate Value Theorem to determine if there is a zero between $$x = 0$$ and $$x = 1.$$

Accepted Answer

Yes, because $$f(0)$$ and $$f(1)$$ have opposite signs.

Question 17

Which of the following is the correct y-intercept for $$f(x) = 2x^3$ - 5x^2$ + 3x - 7$$?

Accepted Answer

$$-7$$

Question 18

If a polynomial function has degree $$n$$, what is the maximum number of real zeros it can have?

Accepted Answer

$$n$$

Polynomial Functions: End Behavior, Zeros, Multiplicity, and Graphing

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