If α and β are the roots of the equation x 2+ x - 1 = 0, then what is the equation whose roots are α 5and β 5?
x2 + 11x - 1 = 0
By Vieta: \(\alpha+\beta=-1,\ \alpha\beta=-1\). Since \(\alpha^2=-\alpha+1\), the power sums \(S_n=\alpha^n+\beta^n\) satisfy \(S_n=-S_{n-1}+S_{n-2}\).
\(S_1=-1,\ S_2=3,\ S_3=-4,\ S_4=7,\ S_5=-11\). Product \(\alpha^5\beta^5=(-1)^5=-1\).
Equation: \(x^2-(-11)x+(-1)=0\Rightarrow\mathbf{x^2+11x-1=0}\).
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