Online Course Discussion Forum

Math Challenge III Algebra 2.29

 
 
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Math Challenge III Algebra 2.29
by James Shi - Saturday, 18 July 2020, 5:42 PM
 

I do not know how to approach this problem. It seems that the LHS is very similar to the expansion of (a - b)^2, in this case, a = z + 1/z and b = z^2 + 1/z^2. I think this has something to do with roots of unity since z and 1/z are conjugates if z is a root of unity. I know there are 14 solutions, but I only have 2: the third roots of unity excluding 1. What strategy should I use to approach this problem? Thank you.

 
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Re: Math Challenge III Algebra 2.29
by Dr. Kevin Wang - Saturday, 18 July 2020, 6:44 PM
 

The equation in question is:

Solve for $z$: \[ \left(z+\frac{1}{z}\right)^2 + \left(z^2+\frac{1}{z^2}\right)^2 - \left(z+\frac{1}{z}\right)\cdot\left(z^2+\frac{1}{z^2}\right)\cdot\left(z^4+\frac{1}{z^4}\right)=3. \]

It is good to expand the left hand side and clear the denominators, and then it is related to $z^{15}+1$.