1.17: Past-paper question 6

4PM1/2R/June/2023 — Question 6 · 10 marks

f(x)=2x2+5x−pf(x) = 2x^2 + 5x - p
The equation f(x)=0\displaystyle f(x) = 0 has roots α\displaystyle \alpha and β\displaystyle \beta
Given that α3+β3=−2158\text{Given that } \alpha^3 + \beta^3 = -\frac{215}{8}
(a) find the value of p\displaystyle p
(5)
Without solving the equation f(x)=0\displaystyle f(x) = 0
(b) form a quadratic equation, with integer coefficients, that has roots
α+βα2andα+ββ2\frac{\alpha + \beta}{\alpha^2} \quad \mathrm{and} \quad \frac{\alpha + \beta}{\beta^2}
(5)