1.2: Roots of a quadratic and a new quadratic equation

4PM1/1R/June/2022 — Question 8 · 10 marks

(a) Using the binomial expansion, or otherwise, find the complete expansion of
(x+y)3.(x+y)^3.
(1)
The quadratic equation
2x2+3x+4=02x^2+3x+4=0
has roots α\displaystyle \alpha and β\displaystyle \beta.
(b) Without solving the equation, find the value of
α3+β3.\alpha^3+\beta^3.
(4)
(c) Hence, form a quadratic equation with integer coefficients that has roots
αβ2andβα2.\frac{\alpha}{\beta^2} \qquad \text{and} \qquad \frac{\beta}{\alpha^2}.
(5)