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Factors, Zeros, and Solutions of Polynomial Equations: Mastery Test $\newline$ $1$ $\newline$ Select the correct answer from each drop-down menu. $\newline$ Consider polynomial function $\newline$ $f$ . $\newline$ $f(x)=(x-1)^{2}(x+3)^{3}(x+1)$ $\newline$ Use the equation to complete each statement about this function. $\newline$ The zero located at $\newline$ $x=1$ has a multiplicity of $\newline$ $\square y$ , and the zero located at $\newline$ $x=-3$ has a multiplicity of $\newline$ $\square$ . $\newline$ The graph of the function will touch, but not cross, the $\newline$ $x$ -axis at an $\newline$ $x$ -value of

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Reflections of functions

Full solution

Q. Factors, Zeros, and Solutions of Polynomial Equations: Mastery Test

\newline

1

\newline

Select the correct answer from each drop-down menu.

\newline

Consider polynomial function

\newline

f

.

\newline

f(x)=(x-1)^{2}(x+3)^{3}(x+1)

\newline

Use the equation to complete each statement about this function.

\newline

The zero located at

\newline

x=1

has a multiplicity of

\newline

\square y

, and the zero located at

\newline

x=-3

has a multiplicity of

\newline

\square

.

\newline

The graph of the function will touch, but not cross, the

\newline

x

-axis at an

\newline

x

-value of

Identify Polynomial Function: The polynomial function $f(x)$ is given as $f(x)=(x-1)^{2}(x+3)^{3}(x+1)$ . To find the multiplicity of the zeros, we need to look at the exponents of the factors in the polynomial. The multiplicity of a zero is the exponent on the factor that produces the zero.
Multiplicity of Zero at $x=1$ : The zero located at $x=1$ comes from the factor $(x-1)$ . Since $(x-1)$ is raised to the power of $2$ , the multiplicity of the zero at $x=1$ is $2$ .
Multiplicity of Zero at $x=-3$ : The zero located at $x=-3$ comes from the factor $(x+3)$ . Since $(x+3)$ is raised to the power of $3$ , the multiplicity of the zero at $x=-3$ is $3$ .
Graph Behavior at Zeros: The behavior of the graph at the zeros depends on the multiplicity of the zeros. If the multiplicity is even, the graph will touch the $x$ -axis at the zero but not cross it. If the multiplicity is odd, the graph will cross the $x$ -axis at the zero.
Multiplicity of Zero at $x=1$ : Since the zero at $x=1$ has a multiplicity of $2$ , which is even, the graph of the function will touch the $x$ -axis at an $x$ -value of $1$ but not cross it.
Multiplicity of Zero at $x=-3$ : Since the zero at $x=-3$ has a multiplicity of $3$ , which is odd, the graph of the function will cross the $x$ -axis at an $x$ -value of $-3$ .

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