The roots of the quadratic x2 - (a + d) x + ad - bc = 0 are α and β. Show that α3 and β3 are the roots of x2 - (a3 + d3 + 3abc + 3bcd) x + (ad - bc)3 = 0.
Solution
We have αβ = ad-bc and α+β = a+d. Hence, α3β3 = (αβ)3 = (ad-bc)3, and α3+β3 = (α+β)3 - 3αβ(α+β) = (a+d)3 - 3(ad-bc)(a+d) = a3+d3 + 3bc(a+d).
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03