Question

Find a polynomial with integer coefficients that satisfies the given conditions.
Q has degree 3 and zeros
−6
and 1 + i.

Answer 1

who can solve it?

Answer 2

Actually, aren't there a family of polynomials that satisfy these conditions? For example if I found a solution, what would stop me from multiplying the whole thing by 2?

Answer 3

Anyway, we know one factor is (x+6), because ((-6)+6) = 0.

Answer 4

If 1+i is a root, then so is 1-i. We thus make factors of them: \[
(x - (1+i))(x-(1-i))(x+6) = 0
\]

Answer 5

It's just a matter of expansion... as I suspected, there are a family of solutions.

Answer 6

\[
\begin{split}
(x - (1+i))(x-(1-i))(x+6) &= (x^2-x-ix-x+1+i+ix-i-i^2)(x+6) \\
&= (x^2-2x+1-i^2)(x+6) \\
&= (x^2-2x+2)(x+6) \\
&= x^3-2x^2+2x + 6x^2 - 12 x + 12 \\
&= x^3+4x^2-10x+12
\end{split}
\]So my final answer is \[ \Large
x^3+4x^2-10x+12
\]

Answer 7

But the complete solution set would be \[ \Large
\forall c\in \mathbb{Z} \quad cx^3+4cx^2-10cx+12c
\]Since any integer coefficients will fly.

Answer 8

thank you WO

Answer 9

WIO*