which of the following describes the roots of the polynomial function (f(x)=(x - 3)^4(x + 6)^2)?\n-3 with…

Explanation:

Step1: Recall root - finding for polynomials

Set (f(x)=(x - 3)^4(x + 6)^2=0).

Step2: Use zero - product property

If (ab = 0), then (a = 0) or (b = 0). So ((x - 3)^4=0) or ((x + 6)^2=0).

Step3: Solve ((x - 3)^4=0)

Taking the fourth - root of both sides, we get (x-3 = 0), so (x = 3). The exponent 4 means the root (x = 3) has multiplicity 4.

Step4: Solve ((x + 6)^2=0)

Taking the square - root of both sides, we get (x+6 = 0), so (x=-6). The exponent 2 means the root (x=-6) has multiplicity 2.

Answer:

3 with multiplicity 4 and -6 with multiplicity 2