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 April 21st, 2012, 09:48 PM #1 Member   Joined: Nov 2011 Posts: 45 Thanks: 0 the fundamental theorem 1) could you show me how you prove that x³ = 8 has three roots? is it (x-0) * (x+0) .... 2) Was this theorem invented after the Complex numbers, or it has a justification of its own? 3) What is the practical, concrete use of the other roots, apart from x=2 ? thanks.
 April 21st, 2012, 10:04 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs Re: the fundamental theorem 1.) Write the equation as: $x^3-2^3=0$ Use the difference of cubes formula: $(x-2)$$x^2+2x+4$$=0$ So, the real root is $x=2$ and the complex conjugate roots we find with the quadratic formula: $x=\frac{-2\pm2i\sqrt{3}}{2}=-1\pm i\sqrt{3}$ 2.) I believe complex numbers were conceived of before the FTOA. 3.) To give one example, without the complex roots many theorems in classical physics governing harmonic motion would be incomplete.
 April 21st, 2012, 11:25 PM #3 Member   Joined: Nov 2011 Posts: 45 Thanks: 0 Re: the fundamental theorem Thanks, Mark. That is how you find the roots. My question is how you prove that a monotonously increasing function has 3 roots? Could you give me a link re point 3? How can the root af any odd power create a problem in physics? Thanks
 April 21st, 2012, 11:31 PM #4 Member   Joined: Nov 2011 Posts: 45 Thanks: 0 Re: the fundamental theorem Oh. I forgot, If I got it right you cannot find the complex roots without finding the real root? is that so? Thanks
 April 22nd, 2012, 01:00 AM #5 Member   Joined: Nov 2011 Posts: 45 Thanks: 0 Re: the fundamental theorem one more question what is i³ ? and sqrt(i)?
 April 22nd, 2012, 01:10 AM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs Re: the fundamental theorem $i^3=i^2\cdot i=-i$ $\sqrt{i}=\frac{1+i}{\sqrt{2}}$
April 22nd, 2012, 04:50 AM   #7
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Re: the fundamental theorem

Quote:
 Originally Posted by MarkFL $i^3=i^2\cdot i=-i$$\sqrt{i}=\frac{1+i}{\sqrt{2}}$
Thanks, can you please specify if you can find the imaginary root without knowing the real one?

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