Second part to this video: http://youtu.be/shEk8sz1oOw
READ FULL DESCRIPTION FOR CLARIFICATIONS, EXTRA INFO, ETC.
If the highest power of a function or polynomial is odd
(e.g.: x^3 or x^5 or x^4371) then it definitely has a solution (or root) among the real numbers. Here's a nice proof demonstrated by Prof David Eisenbud from the Mathematical Sciences Research Institute.
At 10:33 Prof Eisenbud intended to say "no rational roots" rather than "no real roots".
At 2:52 we should have put (2,5) rather than (2,4)
Also, Prof Eisenbud adds that "The Dedekind cut corresponding to the root is: (Rationals x where f(x) is less than or equal to zero) + (Rationals x where f(x) is greater than zero)"
Numberline stuff: http://youtu.be/JmyLeESQWGw
Dedekind cuts: http://en.wikipedia.org/wiki/Dedekind_cut
Numberphile on Facebook: http://www.facebook.com/numberphile
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Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile
Videos by Brady Haran
A run-down of Brady's channels: http://bit.ly/bradychannels
Tagged under: numberphile,polynomial,equation,roots,solution
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