Question

the graph to the right is a complete graph, that is, it is continuous and displays the functions end behavior. all zeros are integers. answer the following questions.

-10,10,1 by -100,100,10

list the zeros whose multiplicity is odd. select the correct choice below and fill in any answer boxes within your choice.
a. -2,3,1 (type an integers or a simplified fractions. use a comma to separate answers as needed.)
b. there are no such zeros.

(b) write an equation, expressed as the product of factors, of a polynomial that the graph might represent. use a leading coefficient of 1 or -1, and make the degree of f as small as possible.
f(x) =
(type your answer in factored form.)

Explanation:

Identify the zeros and their behavior

Using the Graph Behavior at Zeros knowledge point
The graph intersects or touches the x-axis at three integer locations:

• At \(x = -2\), the graph crosses the x-axis.
• At \(x = 1\), the graph crosses the x-axis.
• At \(x = 3\), the graph crosses the x-axis.

Determine multiplicities of the zeros

Using the Multiplicity of Zeros knowledge point

• Since the graph crosses the x-axis at \(x = -2\), \(x = 1\), and \(x = 3\), all three zeros have odd multiplicity.
• To make the degree of \(f\) as small as possible, we assign the smallest odd multiplicity, which is \(1\), to each zero.
• There are no zeros where the graph touches and turns around (even multiplicity).

Determine the leading coefficient

Using the End Behavior of Polynomials knowledge point

• As \(x \to \infty\), \(f(x) \to \infty\).
• As \(x \to -\infty\), \(f(x) \to -\infty\).
• This end behavior indicates an odd-degree polynomial with a positive leading coefficient.
• Since the leading coefficient must be \(1\) or \(-1\), we choose \(a = 1\).

Construct the polynomial equation

Using the Polynomial Functions knowledge point

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

Answer:

Question 1

List the zeros whose multiplicity is even. Select the correct choice below and fill in any answer boxes within your choice.
<blank>There are no such zeros.</blank>

Question 2

List the zeros whose multiplicity is odd. Select the correct choice below and fill in any answer boxes within your choice.
<blank>-2, 1, 3</blank>

Question 3

(b) Write an equation, expressed as the product of factors, of a polynomial that the graph might represent. Use a leading coefficient of 1 or -1, and make the degree of f as small as possible.
\(f(x) =\) <blank>\((x+2)(x-1)(x-3)\)</blank>