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 Website: http://www.numberphile.com/ Numberphile on Facebook: http://www.facebook.com/numberphile Numberphile tweets: https://twitter.com/numberphile Google Plus: http://bit.ly/numberGplus Tumblr: http://numberphile.tumblr.com 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