Mathematics A, Further H235: Further Algebra Topical Past Paper Questions

In this question you must show detailed reasoning. You are given that a is a real root of the equation $ x^{4}+x^{3}+3x^{2}-5x=0 $. You are also given that $ a+2+3i $ is one root of the equation $$ z^{4}-2(1+a)z^{3}+(21a-10)z^{2}+(86-80a)z+(285a-195)=0. $$ Determine all possible…

In this question you must show detailed reasoning. The roots of the equation $ 5x^{2}-3x+12=0 $ are $ \alpha $ and $ \beta $. By considering the symmetric functions of the roots, $ \alpha + \beta $ and $ \alpha\beta $, determine the exact value of $ \frac{1}{\alpha^{2}} + \frac{1…

The equation $ 2x^{3} + 3x^{2} - 2x + 5 = 0 $ has roots $ \alpha, \beta $ and $ \gamma $. Use a substitution to find a cubic equation with integer coefficients whose roots are $ \alpha+1 $, $ \beta+1 $ and $ \gamma+1 $. [4]