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}+6x^{2}-4x - 24
determine the zero(s), if they exist.
the zero(s) is/are - 6, - 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 are three zeros. the multiplicity of the smallest zero is . the multiplicity of the largest zero is . the multiplicity of the other zero is .
(simplify your answers.)
b. there is one zero. the multiplicity of the zero is .
(simplify your answer.)
c. there are two zeros. the multiplicity of the smallest zero is . the multiplicity of the largest zero is .
(simplify your answers.)

Explanation:

Step1: Factor the polynomial

First, group the terms of \(f(x)=x^{3}+6x^{2}-4x - 24\) as \((x^{3}+6x^{2})-(4x + 24)\). Then factor out the greatest - common factor from each group: \(x^{2}(x + 6)-4(x + 6)=(x + 6)(x^{2}-4)\). Further factor \(x^{2}-4\) using the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\), so \(f(x)=(x + 6)(x + 2)(x - 2)\).

Step2: Determine the multiplicities

Since \(f(x)=(x + 6)(x + 2)(x - 2)\), each factor is to the first power. The zeros are \(x=-6,x=-2,x = 2\). For a zero \(c\) of a polynomial \(P(x)\), if \((x - c)^k\) is a factor of \(P(x)\) and \((x - c)^{k+1}\) is not a factor, then the multiplicity of the zero \(c\) is \(k\). Here, \(k = 1\) for \(x=-6,x=-2,x = 2\).

Answer:

A. There are three zeros. The multiplicity of the smallest zero (\(x=-6\)) is \(1\). The multiplicity of the largest zero (\(x = 2\)) is \(1\). The multiplicity of the other zero (\(x=-2\)) is \(1\).