what is a polynomial function in standard form with zeros 1, 2, -2 and -3

The zeros of the polynomial are 1, 2, -2 and -3, so this polynomial must have at least one of each of these factors:

(x-1), (x-2), (x-(-2)), and (x-(-3)); rewriting: (x-1), (x-2), (x+2), and (x+3).

Thus, any such polynomial must have a factor (x-1)(x-2)(x+2)(x+3).

The simplest such polynomial we can think of, is p(x)=(x-1)(x-2)(x+2)(x+3).

To write in standard form, lets first multiply the factors two by two as follows:

[tex](x-2)(x+2)=x^2-4[/tex] by the difference of squares formula, 

[tex](x-1)(x+3)=x^2+2x-3[/tex].

Next, we multiply our results:

[tex](x^2-4)(x^2+2x-3)=x^2(x^2+2x-3)-4(x^2+2x-3)[/tex]

[tex]=x^4+2x^3-3x^2-4x^2-8x+12=x^4+2x^3-7x^2-8x+12[/tex].


Answer: [tex]x^4+2x^3-7x^2-8x+12[/tex]