Question

What would be a polynomial equation with the root 2+i? Please also name another root of this equation. Thank You :)

Answer 1

An example with real coefficients would be:

`x^2-4x+5 = 0`

which also has root `2-i`

Explanation:

If any coefficients are allowed then a polynomial equation of minimum degree with root `2+i` is:

`x - (2+i) = 0`

This linear polynomial equation has no other roots.

Typically what is wanted is not a polynomial with complex coefficients, but one with real coefficients.

If so, then any non-real complex roots will occur in complex conjugate pairs (essentially because `i` and `-i` are indistinguishable from the perspective of the real numbers).

So if `2+i` is one root, then another would be `2-i`.

A polynomial of minimum degree with these zeros would be:

`(x-(2+i))(x-(2-i)) = ((x-2)-i)((x-2)+i)`

`color(white)((x-(2+i))(x-(2-i))) = (x-2)^2-i^2`

`color(white)((x-(2+i))(x-(2-i))) = x^2-4x+4+1`

`color(white)((x-(2+i))(x-(2-i))) = x^2-4x+5`

So we can write a suitable equation as:

`x^2-4x+5=0`