Question

Find a polynomial of degree `3` that has zeros, `1`, `-2`, and `3`, and in which the coefficient of `x^2` is `3`?

Answer 1

`f(x) = -3/2x^3+3x^2+15/2x-9`

Explanation:

Each zero `a` corresponds to a linear factor `(x-a)`.

So we can write a monic cubic polynomial with the required zeros by multiplying as follows:

`(x-1)(x+2)(x-3) = x^3-2x^2-5x+6`

Then to make the coefficient of `x^2` into `3`, multiply by `-3/2` to get:

`f(x) = -3/2x^3+3x^2+15/2x-9`