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 + 8x^2 - 4x - 32
determine the zero(s), if they exist.
the zero(s) is/are - 8, - 2,2. (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.
a. there is one zero. the multiplicity of the zero is

b. there are two zeros. the multiplicity of the smallest zero is. the multiplicity of the largest zero is (simplify your answers.)
c. there are three zeros. the multiplicity of the smallest zero is 1. the multiplicity of the other zero is 1. the multiplicity of the largest zero is 1. (simplify your answers.)
determine the behavior of the function at each zero. select the correct choice below and, if necessary, fill in the answer boxes within your choice.
a. the graph crosses the x - axis at x = - 8. the graph touches the x - axis and turns around at x = - 2. (type integers or decimals. simplify your answers. use a comma to separate answers as needed.)
b. the graph touches the x - axis and turns around at all zeros.
c. the graph crosses the x - axis at all zeros.

Explanation:

Step1: Recall zero - multiplicity rule

If a polynomial \(f(x)\) has a factor \((x - c)^n\), \(c\) is a zero and \(n\) is its multiplicity. If \(n\) is odd, the graph crosses the \(x\) - axis at \(x = c\); if \(n\) is even, the graph touches the \(x\) - axis and turns around at \(x = c\).

Step2: Analyze given zeros and multiplicities

The zeros of \(f(x)=x^{3}+8x^{2}-4x - 32\) are \(x=-8,x = - 2,x = 2\) and each has a multiplicity of \(1\) (since the factors are \((x + 8)(x + 2)(x - 2)\) and the power of each factor is \(1\)).

Step3: Determine graph - axis behavior

Since the multiplicity of each zero (\(-8,-2,2\)) is \(1\) (an odd number), the graph of the function \(y = f(x)\) crosses the \(x\) - axis at \(x=-8,x=-2,x = 2\).

Answer:

C. There are three zeros. The multiplicity of the smallest zero (\(x=-8\)) is \(1\). The multiplicity of the other zero (\(x=-2\)) is \(1\). The multiplicity of the largest zero (\(x = 2\)) is \(1\).
A. The graph crosses the \(x\) - axis at \(x=-8\), \(x=-2\), \(x = 2\).