Question

find the zeros for the polynomial function and give the multiplicity for each zero. state whether the graph crosses the x - axis or touches the x - axis and turns around at each zero.
$f(x)=x^{3}+7x^{2}-16x - 112$

determine the zero(s), if they exist.
the zero(s) is/are \\(\square\\).
(type integers or decimals. use a comma to separate answers as needed.)
determine the multiplicities of the zero(s), if they exist. select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
\\(\bigcirc\\) a. there are two zeros. the multiplicity of the smallest zero is \\(\square\\). the multiplicity of the largest zero is \\(\square\\).
(simplify your answers.)
\\(\bigcirc\\) b. there is one zero. the multiplicity of the zero is \\(\square\\).
(simplify your answer.)
\\(\bigcirc\\) c. there are three zeros. the multiplicity of the smallest zero is \\(\square\\). the multiplicity of the largest zero is \\(\square\\). the multiplicity of the other zero is \\(\square\\).
(simplify your answers.)

Explanation:

Step1: Factor the polynomial by grouping

Given \( f(x) = x^3 + 7x^2 - 16x - 112 \), group the terms:
\( (x^3 + 7x^2) + (-16x - 112) \)
Factor out the GCF from each group:
\( x^2(x + 7) - 16(x + 7) \)
Now factor out \( (x + 7) \):
\( (x + 7)(x^2 - 16) \)
Notice that \( x^2 - 16 \) is a difference of squares, so factor it further:
\( (x + 7)(x - 4)(x + 4) \)

Step2: Find the zeros

To find the zeros, set \( f(x) = 0 \):
\( (x + 7)(x - 4)(x + 4) = 0 \)
Using the zero - product property, we have:
\( x + 7 = 0 \) or \( x - 4 = 0 \) or \( x + 4 = 0 \)
Solving for \( x \):
\( x=-7 \), \( x = 4 \), \( x=-4 \)

Step3: Determine the multiplicities

For a factored polynomial of the form \( f(x)=(x - a)^n(x - b)^m(x - c)^p\), the multiplicity of the zero \( a \) is \( n \), the multiplicity of the zero \( b \) is \( m \), and the multiplicity of the zero \( c \) is \( p \).
In our factored form \( f(x)=(x + 7)^1(x - 4)^1(x + 4)^1 \), each factor has an exponent of 1. So the multiplicity of \( x=-7 \), \( x = 4 \), and \( x=-4 \) is 1.

For the zeros:

The zeros are \( -7, - 4,4 \)

For the multiplicities:

We have three zeros. The multiplicity of the smallest zero (\( x=-7 \)) is 1. The multiplicity of the largest zero (\( x = 4 \)) is 1. The multiplicity of the other zero (\( x=-4 \)) is 1.

Answer:

The zero(s) is/are \(-7, - 4,4\)

For the multiplicities, the correct choice is C. The multiplicity of the smallest zero (\(-7\)) is \(1\). The multiplicity of the largest zero (\(4\)) is \(1\). The multiplicity of the other zero (\(-4\)) is \(1\)