How do you know that a cubic equation with real coefficients must have at least one real root?

Question

anonymous · Answer

u just have to know :P

anonymous · Answer

it's like a leap of faith

phi · Answer

a cubic always has three roots.
complex roots always come in pairs (complex conjugates)
so your only choices are:  3 real roots
or 2 complex (which will be conjugates) plus 1 real.
3 complex not allowed (they come in pairs)

dumbcow · Answer

agreed
Also graphically a cubic must either continue to increase or decrease indefinitely so it must cross the x-axis at least once

anonymous · Answer

why do complex roots come in pairs?

dumbcow · Answer

comes from definition of imaginary number
$$i^{2} = -1$$
$$\rightarrow \pm \sqrt{-1} = \pm i$$
so there is a +- component to every complex root

dumbcow · Answer

sorry its not the best explanation

anonymous · Answer

its fine. im just a bit rusty for because my pre calc class ended 6 months ago and now we're jumping straight into calc

dumbcow · Answer

oh good luck i'm sure you'll do fine
there will be plenty of review im sure