Question

How do you find a third degree polynomial given roots `-1` and `3+i`?

Answer 1

The third-degree polynomial with roots `-1` and `3 + i` is
`f(x) = x^3 - 5x^2 + 4x + 10`.

Explanation:

Complex roots are always in pairs, and the pairs are conjugates. Therefore, even though you are only given two roots, the polynomial actually has three roots. The third root is `3 - i`.

Remember that a root is represented by `k`, and that the factor which yields a root is in the form `x - k`. Therefore, to write the polynomial which has the given roots and a leading coefficient of `1`, simply set up the roots in factor form and multiply them.

`f(x) = (x - -1)(x - [3 + i])(x - [3 - i])`
`f(x) = (x + 1)(x - 3 - i)(x - 3 + i)`
`f(x) = (x + 1)(x^2 - 3x +ix - 3x + 9 - 3i - ix + 3i - i^2)`
`f(x) = (x + 1)(x^2 - 6x + 9 +1)`
`f(x) = (x + 1)(x^2 - 6x + 10)`
`f(x) = x^3 - 6x^2 + 10x + x^2 - 6x + 10`
`f(x) = x^3 - 5x^2 + 4x + 10`