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 + 4x^2 - 9x - 36
(type a whole number.)
the multiplicity of the second greatest zero is 1.
(type a whole number.)
the multiplicity of the smallest zero is 1.
(type a whole number.)
determine whether the graph crosses the x - axis, or touches the x - axis and turns around at each x - intercept
what is the behavior of the graph at the greatest zero?
crosses the x - axis
touches the x - axis and turns around
what is the behavior of the graph at the second greatest zero?
crosses the x - axis
touches the x - axis and turns around

Explanation:

Step1: Factor the polynomial by grouping

Group the terms: $f(x)=(x^{3}+4x^{2})-(9x + 36)$. Then factor out the greatest - common factor from each group: $x^{2}(x + 4)-9(x + 4)=(x + 4)(x^{2}-9)$.

Step2: Further factor the difference of squares

Since $x^{2}-9=(x + 3)(x - 3)$ (using the formula $a^{2}-b^{2}=(a + b)(a - b)$ where $a=x$ and $b = 3$), then $f(x)=(x + 4)(x+3)(x - 3)$.

Step3: Find the zeros

Set $f(x)=0$. Then $(x + 4)(x + 3)(x - 3)=0$. By the zero - product property, $x+4=0$ gives $x=-4$, $x + 3=0$ gives $x=-3$, and $x - 3=0$ gives $x = 3$.

Step4: Determine the multiplicity and graph behavior

The zeros are $x=-4,x=-3,x = 3$. Each factor is to the first power, so the multiplicity of each zero is 1. When the multiplicity of a zero is 1, the graph of the function crosses the x - axis at that zero.

Answer:

The zeros are $x=-4,x=-3,x = 3$ with multiplicity 1 for each zero. The graph crosses the x - axis at $x=-4$, crosses the x - axis at $x=-3$, and crosses the x - axis at $x = 3$.