Question

Given a degree 5 polynomial with real coefficients has roots 3,2-i,6+i, find the other missing roots?

Answer 1

I got 2+i and 6-i as the other missing roots am i correct?

Answer 2

that's correct, good work

Answer 3

ok because i recalled that law which you said that a-bi can be a+bi something like that correct me if i am wrong.

Answer 4

yes, that's the conjugate root theorem

Answer 5

alright thank you.

Answer 6

awesome teaching !!

Answer 7

easy to say with hindsight:

\((x - z_0 ) ( x - \bar z_0) = x^2 - x (\bar z_0 + z_0) + z_0 \bar z_0 = x^2 - 2 x_0 x + |z_0|^2\)

All of which follows from the fact that \(z_0 + \bar z_0 = 2 x_0\) and \(z_0 \bar z_0 = |z_0|^2\)

So:

\((x - \alpha)(x - \beta)(x - z_0 ) ( x - \bar z_0) = (x - \alpha)(x - \beta) ( x^2 - 2 x_0 x + |z_0|^2)\)