Question

How do you write a polynomial in standard form given the zeros 3, 1, 2, and-3?

Answer 1

`f(x) = x^4-3x^3-7x^2+27x-18`

Explanation:

Using `x` as our variable name, each zero `a` corresponds to a factor `(x-a)`.

So we can write:

`f(x) = (x-3)(x-1)(x-2)(x+3)`

`color(white)(f(x)) = ((x-3)(x+3))((x-1)(x-2))`

`color(white)(f(x)) = (x^2-9)(x^2-3x+2)`

`color(white)(f(x)) = x^2(x^2-3x+2)-9(x^2-3x+2)`

`color(white)(f(x)) = (x^4-3x^3+2x^2)-(9x^2-27x+18)`

`color(white)(f(x)) = x^4-3x^3-7x^2+27x-18`

This is the simplest polynomial with the given zeros. Any polynomial in `x` with these zeros is a multiple (scalar or polynomial) of this `f(x)`.