need help with math please?

Question

what do they mean by "write a polynomial function in standard form with the given zeros and their multiplicities"

and explain how you would find the answer for
"1( multiplicity 2), -2 (multiplicity 3)"

Answer 1

A polynomial has a zero at x=a when one of its factors is (x - a).

Multiplicity refers to the number of times the given root appears in the equation. So, if one says "a zero at 3 with multiplicity 4," it means that (x - 3)^4 is a factor of the function.

"1( multiplicity 2), -2 (multiplicity 3)"

Means that the term having a zero at x=1 is squared, and the term having a zero at x=-2 is cubed. This is written as:

(x - 1)^2 * (x + 2)^3

All that remains is to multiply it out to put it in standard form:

(x^2 - 2x + 1) * (x + 2) * (x^2 + 4x + 4)

(x^2 - 2x + 1) * (x^3 + 6x^2 + 12x + 8)

x^5 + 6x^4 + 12x^3 + 8x^2 - 2x^4 -12x^3 - 24x^2 - 16x + x^3 + 6x^2 + 12x + 8

x^5 + 4x^4 + x^3 - 10x^2 - 4x + 8

Answer 2

multiplicity is the number of times a particular number is a root, for examples

1 is a root of
(x-1) with multiplicity 1
(x-1)^2 with multiplicity 2
(x-1)^5 with multiplicity 5

to answer your question:
the polynomial is
(x-1)^2*(x+2)^3

expand this and you have your polynomial in standard form

(x^2 - 2x +1) (x^3 + 6x^2 + 12 x + 8) =
x^5 + 4x^4 + x^3 -10x^2 - 4x +8