Which of the following describes the roots of the polynomial function f(x)=(x-3)^4(x-6)^2? –3 with multiplicity 2 and 6 with multiplicity 4 –3 with multiplicity 4 and 6 with multiplicity 2 3 with multiplicity 2 and –6 with multiplicity 4 3 with multiplicity 4 and –6 with multiplicity 2

You made a mistake either on the polynomial function or on the choices.

I will explain you how to get the answer, using the same polynomial given:

Answer: 3 with multiplicity 2 and 6 with multiplicity 3.

The given polynomial function  is f(x)=(x-3)^4 * (x-6)^2

or written with the math tool:

 [tex]f(x)=(x-3)^4(x-6)^2[/tex]

The roots are the x-intercepts of the function. This isthe values of x for which the function is zero. Thus, you have to find the solution to this equation:

[tex](x-3)^2(x-6)^2[/tex]

The solutions are:

1)  (x-3)^4 = 0 => (x-3)=0 => x = 3. The multiplicity of the root is the exponent of the factor, so the multiplicity of this root is 4.

You can understand that if you think that (x-3)^4 = (x-3)(x-3)(x-3)(x-3), which tells you that (x - 3) is 0 for each one of the 4 parenthesis.

2) (x - 6)^2 = 0 => x = 6, with multiplicity 2 (the exponent of the factor x - 6).

Then, the answer is 3 with multiplicity 4 and 6 with multiplicity 2.