1.29: Past-paper question 7

4PM1/1/November/2025 — Question 7 · 10 marks

(a) Show that (α+β)(α2−αβ+β2)=α3+β3\displaystyle (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) = \alpha^3 + \beta^3
(1)
The equation 2x2+8x−k=0\displaystyle 2x^2 + 8x - k = 0 has roots α\displaystyle \alpha and β\displaystyle \beta and where k\displaystyle k is a constant
Given that α3+β3=−94\displaystyle \alpha^3 + \beta^3 = -94
(b) show that k=5\displaystyle k = 5
(4)
Given that α>β\displaystyle \alpha > \beta and without solving the equation 2x2+8x−k=0\displaystyle 2x^2 + 8x - k = 0
(c) (i) show that α−β=26\displaystyle \alpha - \beta = \sqrt{26}
(3)
(ii) hence find the exact value of α3−β3\displaystyle \alpha^3 - \beta^3
(2)