Factoring A Cubic Function

Question

Can someone help me factor $\;2x^3-1=0\;$ ? I've never known how to factor cubic functions. There should be cubic roots involved.

Answer 1

This equation is quite simple to factor completely as it involves the cube roots (slightly modified) of unity. The three cube roots of unity are (by inspection): 1, $-\frac1 2+i\sqrt3/2$, $-\frac1 2-i\sqrt3/2$, sometimes denoted: 1, $\omega_1, \omega_2$. (You should do some simple (complex) algebra to verify that when each of these is cubed it produces the result of 1. The roots of your equation are the same but multiplied by the cube-root of 2.

Answer 2

Hint: Set $u=\sqrt[3]{2}x$. Then $2x^3-1=u^3-1=(u-1)(u^2+u+1)$.