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

Step-by-step 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.