Why do roots come in conjugate pairs?

Power Overwhelming

This is an expanded version of an answer I gave to a question that came up while I was assisting the 2014-2015 WOOT class. It struck me as an unusually good way to motivate higher math using stuff that people notice in high school but for some reason decide to not think about.

In high school precalculus, you’ll often be asked to find the roots of some polynomial with integer coefficients. For instance,

$latex displaystyle x^3 – x^2 – x – 15 = (x-3)(x^2+2x=5) &fg=000000$

has roots $latex {3}&fg=000000$, $latex {1+2i}&fg=000000$, $latex {-1-2i}&fg=000000$. Or as another example,

$latex displaystyle x^3 – 3x^2 – 2x + 2 = (x+1)(x^2-4x+2) &fg=000000$

has roots $latex {-1}&fg=000000$, $latex {2 + sqrt 2}&fg=000000$, $latex {2 – sqrt 2}&fg=000000$. You’ll notice that the “weird” roots, like $latex {1 pm 2i}&fg=000000$ and $latex {2 pm sqrt 2}&fg=000000$, are coming up in pairs. In fact, I think…

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