Question

How do you find a polynomial of degree 3 that has zeros of -3, 0, 1?

Answer 1

`P(x)=x^3+2x^2-3x`

Explanation:

A polynomial has `alpha` as a zero if and only if `(x-alpha)` is a factor of the polynomial. Working backwards, then, we can generate a polynomial with any zeros we desire by multiplying such factors.

We want a polynomial `P(x)` with zeros `-3, 0, 1`, so:

`P(x) = (x-(-3))(x-0)(x-1)`

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

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

`=x(x^2+2x-3)`

`=x^3+2x^2-3x`

Note that we could also multiply by any nonzero constant without changing the zeros, if a different polynomial is desired.