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 1. the multiplicity of the smallest zero is 2. (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.)
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 touches the x - axis and turns around at all zeros.
b. the graph crosses the x - axis at x = and touches the x - axis and turns around at x =.
(simplify your answers. type integers or decimals. use a comma to separate answers as needed.)
c. the graph crosses the x - axis at all zeros.

Explanation:

Step1: Recall zero - finding rule

Set \(f(x)=0\), so \(-4(x + 3)(x + 4)^2=0\). By the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b=0\). So \(x+3 = 0\) gives \(x=-3\) and \((x + 4)^2=0\) gives \(x=-4\).

Step2: Determine multiplicities

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

Step3: Determine graph behavior

If the multiplicity of a zero is odd, the graph of the polynomial function crosses the \(x\) - axis at that zero. If the multiplicity is even, the graph touches the \(x\) - axis and turns around at that zero. Since the multiplicity of \(x=-3\) is 1 (odd), the graph crosses the \(x\) - axis at \(x=-3\). Since the multiplicity of \(x=-4\) is 2 (even), the graph touches the \(x\) - axis and turns around at \(x=-4\).

Answer:

• For the multiplicities: A. There are two zeros. The multiplicity of the largest zero is 1. The multiplicity of the smallest zero is 2.
• For the graph behavior: B. The graph crosses the \(x\) - axis at \(x=-3\) and touches the \(x\) - axis and turns around at \(x=-4\).