Question

Which of the following describes the roots of the polynomial function f(x) = (x-3)^4 (x + 6) squared?
–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

Answer 1

Option d: 3 with multiplicity 4 and –6 with multiplicity 2

Explanation:

The function is $f(x)=(x-3)^4(x+6)^2$

To determine the roots of a polynomial, let us substitute $f(x)=0$ in the function $f(x)=(x-3)^4(x+6)^2$

Thus, we have,

$0=(x-3)^{4}(x+6)^{2}$

Thus, we have,

$0=(x-3)^{4} \ and\ 0=(x+6)^{2}$

First solving the expression $0=(x-3)^{4}$, we have,

$x=3$ with multiplicity 4.

Also, solving the expression $0=(x+6)^{2}$, we have,

$x=-6$ with multiplicity 2.

Thus, the roots of the polynomial function is 3 with multiplicity 4 and –6 with multiplicity 2

Hence, Option d is the correct answer.