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- Feb 14, 2012

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Problem:

The real numbers $\displaystyle \alpha$, $\displaystyle \beta$ satisfy the equations

$\displaystyle \alpha^3-3\alpha^2+5\alpha-17=0,$

$ \displaystyle \beta^3-3\beta^2+5\beta+11=0.$

Find $\displaystyle \alpha + \beta$.

This problem has me stumped. It's not that I didn't try, believe you me, but I kept returning back to square one no matter how hard I tried. It's really annoying and I don't believe the author wants us to use the cubic formula to find each root explicitly (or even a numeric root-finding technique) to get the sum, but at this moment, I just don't see any way to manipulate the given two cubic polynomials to find the desired sum.

Could you please shed some light on this problem for me?

Thanks in advance.