Q: A polynomial $$f(x)$$ has zeros at $$x = -6$$ and $$x = -4$$, both with multiplicity 1. If the degree is even and the leading coefficient is positive, which of the following best describes the end behavior of $$f(x)$$?

As $$x \to \pm\infty$$, $$f(x) \to \infty.$$

Q: If a polynomial $$f(x)$$ has an even degree and a negative leading coefficient, what is the end behavior of $$f(x)$$?

As $$x \to \pm\infty$$, $$f(x) \to -\infty.$$

Question 1

Given the polynomial $$f(x) = (x - 4)^2$ (x + 4)^3$, what are the zeros and their respective multiplicities?

Accepted Answer

Zero at $$x = 4$$ with multiplicity 2; zero at $$x = -4$$ with multiplicity 3.

Question 2

For the polynomial $$f(x) = 2(x + 3)(x + 5)^2$(x - 3)^2$, which of the following lists all zeros and their multiplicities correctly?

Accepted Answer

Zero at $$x = -3 ($$multiplicity 1), $$x = -5 ($$multiplicity 2), $$x = 3 ($$multiplicity 2).

Question 3

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

Accepted Answer

Degree $$7$$

Question 4

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

Accepted Answer

The graph touches the $$x-$$axis and turns around at $$x = 4.$$

Question 5

Given the polynomial $$f(x) = 7(x + 5)(x + 3)^2$(x - 1)^2$, what is the sign of f(x$$)$$ for x$$ \u003c -5$$?

Accepted Answer

$$f(x) \u003c 0$$

Question 6

For the polynomial $$g(x) = -2(x + 4)(x - 2)(x - 4)^2$, what is the sign of g(x$$)$$ for x$$ \u003e 4$$?

Accepted Answer

$$g(x) \u003c 0$$

Question 7

A polynomial $$f(x)$$ has zeros at $$x = -8, -4, 0, 4. $$The zero at $$x = -8$$ has even multiplicity, $$x = -4$$ has odd multiplicity, $$x = 0$$ has odd multiplicity, and $$x = 4$$ has even multiplicity. What is the minimum possible degree of $$f(x)$$?

Accepted Answer

Degree $$6$$

Question 8

Refer to the first graph provided. Which of the following statements is true about the leading coefficient and degree of the polynomial?

Accepted Answer

The degree is even and the leading coefficient is positive.

Question 9

Given the polynomial $$f(x) = -2(x + 3)^3$ x^4$ (x - 2)(x - 5)^2$, what is the sign of f(x$$)$$ for x$$ \u003c -3$$?

Accepted Answer

$$f(x) \u003c 0$$

Question 10

For the polynomial $$g(x) = 5x^3$(x - 1)^4$(x - 2)(x - 4)^2$, what is the sign of g(x$$)$$ for x$$ \u003e 4$$?

Accepted Answer

$$g(x) \u003e 0$$

Question 11

A chemical reaction is modeled by the polynomial $$f(x) = (x - 2)^3$(x + 1). If$$ $$x$$ represents the concentration of a reactant, what does the zero at $$x = 2$$ with multiplicity 3 indicate about the reaction?

Accepted Answer

The reaction rate changes direction at $$x = 2$$ and the reactant concentration is a triple root.

Question 12

A polynomial $$f(x)$$ has zeros at $$x = -6$$ and $$x = -4$$, both with multiplicity 1. If the degree is even and the leading coefficient is positive, which of the following best describes the end behavior of $$f(x)$$?

Accepted Answer

As $$x 	o \pm\infty$$, $$f(x) 	o \infty.$$

Question 13

Refer to the second graph provided. Which of the following is a possible formula for the polynomial of least degree that fits the graph?

Accepted Answer

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

Question 14

Refer to the third graph provided. Which of the following is a possible formula for the polynomial of least degree that fits the graph?

Accepted Answer

$$f(x) = \frac{1}{10}(x + 2)(x - 3)^3$

Question 15

If a polynomial $$f(x)$$ has a zero at $$x = 0$$ with multiplicity 1, what does the graph do at $$x = 0$$?

Accepted Answer

The graph crosses the $$x-$$axis at $$x = 0.$$

Question 16

A polynomial $$f(x)$$ has zeros at $$x = 2 ($$multiplicity 2), $$x = -1 ($$multiplicity 3), and $$x = 4 ($$multiplicity 1). What is the minimum possible degree of $$f(x)$$?

Accepted Answer

Degree $$6$$

Question 17

If a polynomial $$f(x)$$ has an even degree and a negative leading coefficient, what is the end behavior of $$f(x)$$?

Accepted Answer

As $$x 	o \pm\infty$$, $$f(x) 	o -\infty.$$

Polynomial Functions: Zeros, Multiplicity, Degree, and Graphs

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