How do I find the roots from this?

Question

Problem: We are give that $2+7i$ is a root of $x^4 - 8x^3 + 73x^2 - 228x + 212=0$

I got stuck on this one: how do I find all the roots? $\left(i=\sqrt{-1}\right)$

Answer 1

So the roots come in complex conjugate pairs (all the coefficients are real numbers, indeed integers). You will have a factor

$\left(x-(2+7i)\right)\left(x-(2-7i)\right)=\left((x-2)+7i\right)\left((x-2)-7i\right)=(x-2)^2+49=x^2-4x+53$

If you divide your original equation through by this factor, you will get another quadratic factor whose roots you can find in the usual way.

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Since you know you have a factor, you don't have to do all the division - see if you can spot some short cuts, which will save a lot of time.