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 by grouping

Group terms of the polynomial:
$$f(x) = (x^3 + 7x^2) + (-16x - 112)$$
Factor out common terms from each group:
$$f(x) = x^2(x + 7) - 16(x + 7)$$
Factor out $(x+7)$:
$$f(x) = (x + 7)(x^2 - 16)$$

Step2: Factor difference of squares

Rewrite $x^2-16$ as a difference of squares:
$$x^2 - 16 = x^2 - 4^2$$
Factor using $a^2-b^2=(a-b)(a+b)$:
$$f(x) = (x + 7)(x - 4)(x + 4)$$

Step3: Find zeros by setting $f(x)=0$

Set each factor equal to 0:

1. $x + 7 = 0 \implies x = -7$
2. $x - 4 = 0 \implies x = 4$
3. $x + 4 = 0 \implies x = -4$

Step4: Determine multiplicities

Each linear factor has an exponent of 1, so each zero has multiplicity 1. For zeros with odd multiplicity, the graph crosses the x-axis.

Answer:

The zero(s) is/are $\boldsymbol{-7, -4, 4}$

For multiplicities:
C. There are three zeros. The multiplicity of the smallest zero is $\boldsymbol{1}$. The multiplicity of the largest zero is $\boldsymbol{1}$. The multiplicity of the other zero is $\boldsymbol{1}$.

At $x=-7$: the graph crosses the x-axis
At $x=-4$: the graph crosses the x-axis
At $x=4$: the graph crosses the x-axis