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)=-4(x + 3)(x + 4)^2
determine the zero(s).
the zero(s) is/are - 3, - 4.
(type integers or decimals. use a comma to separate answers as needed.)
determine the multiplicities of the zero(s). select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
a. there are two zeros. the multiplicity of the largest zero is. the multiplicity of the smallest zero is.
(simplify your answers.)
b. there is one zero. the multiplicity of the zero is.
(simplify your answer.)
c. there are three zeros. the multiplicity of the largest zero is. the multiplicity of the smallest zero is. the multiplicity of the other zero is.
(simplify your answers.)

Explanation:

Step1: Recall zero - multiplicity concept

The zeros of a polynomial function \(y = f(x)\) are the values of \(x\) for which \(f(x)=0\). If \((x - a)^n\) is a factor of the polynomial, then \(a\) is a zero of the polynomial and \(n\) is its multiplicity.
For the polynomial \(f(x)=-4(x + 3)(x + 4)^2\), we set \(f(x)=0\).

Step2: Find the zeros

Set each factor equal to zero:
\(x+3 = 0\) gives \(x=-3\) and \((x + 4)^2=0\) gives \(x=-4\).

Step3: Determine multiplicities

For the factor \((x + 3)\), the exponent of \((x + 3)\) is \(1\), so the multiplicity of the zero \(x=-3\) is \(1\).
For the factor \((x + 4)^2\), the exponent is \(2\), so the multiplicity of the zero \(x=-4\) is \(2\). Since \(-3>-4\), the largest zero is \(x=-3\) with multiplicity \(1\) and the smallest zero is \(x = - 4\) with multiplicity \(2\).

Answer:

A. There are two zeros. The multiplicity of the largest zero is \(1\). The multiplicity of the smallest zero is \(2\).