Question

A polynomial function has a root of –6 with multiplicity 1, a root of –2 with multiplicity 3, a root of 0 with multiplicity 2, and a root of 4 with multiplicity 3. If the function has a positive leading coefficient and is of odd degree, which statement about the graph is true?
The graph of the function is positive on (–6, –2).
The graph of the function is negative on (mc025-1.jpg, 0).
The graph of the function is positive on (–2, 4).
The graph of the function is negative on (4, mc025-2.jpg).

Answer 1

In order to find the roots we will set the polynomials equal to zero. They will make the values of x function equal to zero.

Root of -6 with multiplicity 1 is the factor (x + 6).

Root of -2 with multiplicity 3 is the factor (x + 2)^3.

Root of 0 with multiplicity 2 is the factor x^2.

Root of 4 with multiplicity 3 is the factor (x - 4)^3.

The function here is

f(x) = x^2(x + 6)(x + 2)^3(x - 4)^3

 

f(x) = x*x(x + 6)(x + 2)(x + 2)(x + 2)(x - 4)(x - 4)(x - 4)

Now if we expand it we will get x^9 and the last term of polynomial will be -3072.