Following a polynomial, find

a. the zero's and the multiplicity of each
b. where the graph crosses or touches the x-axis
c. the number of turning points
d. the end behavior

f(x)=(x+3)(x-5)(x+5)

To find the zeros and multiplicities of a polynomial, we need to factor it and set each factor equal to zero.

a) Zeros and Multiplicities:
Given function: f(x) = (x+3)(x-5)(x+5)

To find the zeros, set each factor equal to zero:
x + 3 = 0 --> x = -3 (Zero with multiplicity 1)
x - 5 = 0 --> x = 5 (Zero with multiplicity 1)
x + 5 = 0 --> x = -5 (Zero with multiplicity 1)

Therefore, the zeros of f(x) are -3, 5, and -5, each with a multiplicity of 1.

b) Where the graph crosses or touches the x-axis:
In the case of this polynomial, since the multiplicity of each zero is 1, the graph will cross the x-axis at each zero. So, the graph of f(x) will cross the x-axis at x = -3, x = 5, and x = -5.

c) Number of turning points:
To find the number of turning points, we need to determine the degree of the polynomial. The degree is equal to the highest power of the variable x. In this case, the degree is 3, as each factor has a power of 1.

For polynomials of odd degree, the number of turning points is one less than the degree. So, for this polynomial, there will be 2 turning points.

d) End behavior:
To determine the end behavior of the polynomial, look at the leading term. In this case, the leading term is x^3, and the coefficient is positive.

For polynomials with positive leading coefficients and odd degrees, the end behavior is as follows:
- As x approaches negative infinity, f(x) approaches negative infinity.
- As x approaches positive infinity, f(x) approaches positive infinity.

So, the end behavior of f(x) = (x+3)(x-5)(x+5) is:
- As x approaches negative infinity, f(x) approaches negative infinity.
- As x approaches positive infinity, f(x) approaches positive infinity.