Question

Find a polynomial function of lowest degree with rational coefficients that has the given numbers as some of its zeros.

1 + i, 3

Answer 1

`f(x) = x^(3) - 5x^(2)   + 7x - 3`

Explanation:

If the polynomial has one root as 1 + i, it's conjugate 1 - i will also be its root.

Therefore, the polynomial has roots 1 + i, 1 - i, and 3.

So, the polynomial function of lowest degree with rational coefficients will be

f (x) = (x - 3)(x - 1 - i)(x - 1 + i)

⇒
`f(x) = (x - 3)[ x^(2) - 2x + (1 + i)(1 - i)]`

⇒
`f(x) = (x - 3)(x^(2) - 2x + 2)`

⇒
`f(x) = x^(3) - 5x^(2)   + 7x - 3` (Answer)