F.4-5___Maths

Sum of roots = -b/a = -(-2) = 2
-> A+B = 2
Product of roots = c/a = -4
-> AB = -4

a)
(A/B) + (B/A)
= A*A / AB + B*B / AB
= (A^2+B^2) / AB
= (A^2+2AB+B^2 - 2AB) / AB
= [ (A+B)^2 - 2AB ] / AB
= [ 2^2 - 2(-4) ] / (-4)
= (4+8) / -4
= 12 / -4
= -3

b)
A^4 + B^4
= A^4 + 2A^2B^2 + B^4 - 2A^2B^2
= (A^2+B^2)^2 - 2(AB)^2
= (A^2 + 2AB + B^2 - 2AB)^2 - 2(AB)^2
= [ (A+B)^2 - 2AB ]^2 - 2(AB)^2
= [ 2^2 - 2(-4) ]^2 - 2(-4)^2
= (4+8)^2 - 32
= 144 - 32
= 112