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

which of the following describes the roots of the polynomial function (f(x)=(x - 3)^{4}(x + 6)^{2})?\n-3 with multiplicity 2 and 6 with multiplicity 4\n-3 with multiplicity 4 and 6 with multiplicity 2\n3 with multiplicity 2 and -6 with multiplicity 4\n3 with multiplicity 4 and -6 with multiplicity 2

which of the following describes the roots of the polynomial function (f(x)=(x - 3)^{4}(x + 6)^{2})?\n-3 with multiplicity 2 and 6 with multiplicity 4\n-3 with multiplicity 4 and 6 with multiplicity 2\n3 with multiplicity 2 and -6 with multiplicity 4\n3 with multiplicity 4 and -6 with multiplicity 2

Answer

Explanation:

Step1: Recall root - finding rule

Set each factor equal to zero. For $(x - 3)^4$, we solve $x-3 = 0$, getting $x = 3$. For $(x + 6)^2$, we solve $x+6=0$, getting $x=-6$.

Step2: Determine multiplicities

The exponent of a factor gives the multiplicity of the corresponding root. The exponent of $(x - 3)$ is 4, so the multiplicity of $x = 3$ is 4. The exponent of $(x + 6)$ is 2, so the multiplicity of $x=-6$ is 2.

Answer:

D. 3 with multiplicity 4 and -6 with multiplicity 2