document.write( "Question 694174: find all 12 solutions to x^12=1
\n" ); document.write( "Already know 1 and -1 are solutions, but I'm not sure how to factor the equation. \n" ); document.write( "

Algebra.Com's Answer #427783 by stanbon(75887) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Complex_Numbers/694174 (2012-12-13 21:23:13): find all 12 solutions to x^12=1
\n" ); document.write( "Already know 1 and -1 are solutions, but I'm not sure how to factor the equation.
\n" ); document.write( "-----
\n" ); document.write( "x^12-1 = 0
\n" ); document.write( "-----
\n" ); document.write( "((x^4)^3-1) = 1
\n" ); document.write( "----
\n" ); document.write( "(x^4-1)((x^4)^2 + x^4 + 1) = 0
\n" ); document.write( "---
\n" ); document.write( "(x^2+1)(x^2-1)((x^4)^2+x^4+1) = 0
\n" ); document.write( "-----
\n" ); document.write( "(x+i)(x-i)(x-1)(x+1)(factors of the quadratic) = 0
\n" ); document.write( "At this point you have zeroes at -i,i,1,-1
\n" ); document.write( "------
\n" ); document.write( "Working with the quadratic:
\n" ); document.write( "x^4 = [-1 +- sqrt(1-4*1)]/2
\n" ); document.write( "---
\n" ); document.write( "x^4 = [(-1/2) +- (i/2)sqrt(3)]
\n" ); document.write( "----
\n" ); document.write( "That gives you an additional 8 roots
\n" ); document.write( "-----
\n" ); document.write( "Comment: Normally you would use DeMoivre's Method to
\n" ); document.write( "find the 12 roots.
\n" ); document.write( "x = cis[(90+360n)/12] where n = 0,1,2,3,4,....,10,11
\n" ); document.write( "--------------------
\n" ); document.write( "Cheers,
\n" ); document.write( "Stan H. \n" ); document.write( "

\n" );