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)=-9(x - 7)(x + 6)^3
determine the zero(s).
the zero(s) is/are 7, - 6.
(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 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.)
c. there is one zero. the multiplicity of the zero is .
(simplify your answer.)

Explanation:

Step1: Identify zero - multiplicity rule

The multiplicity of a zero \(r\) of a polynomial \(P(x)=(x - r)^n\cdot g(x)\) is \(n\).

Step2: Analyze the given polynomial

For \(f(x)=-9(x - 7)(x + 6)^3\), when \(x-7 = 0\), \(x = 7\) and its multiplicity is \(1\) since the power of \((x - 7)\) is \(1\). When \(x+6=0\), \(x=-6\) and its multiplicity is \(3\) since the power of \((x + 6)\) is \(3\).

Step3: Determine graph - behavior rule

If the multiplicity \(n\) of a zero is odd, the graph of the polynomial crosses the \(x\) - axis at that zero. If \(n\) is even, the graph touches the \(x\) - axis and turns around. For \(x = 7\) (multiplicity \(1\), odd), the graph crosses the \(x\) - axis. For \(x=-6\) (multiplicity \(3\), odd), the graph crosses the \(x\) - axis.

Answer:

A. There are two zeros. The multiplicity of the largest zero (\(7\)) is \(1\). The multiplicity of the smallest zero (\(-6\)) is \(3\).