f.4 a.maths (10 points!!)

sum of roots:α+β=-b
product of roots:αβ=c
α4 +β4=(α+β)4-4α3β-6α2β2-4αβ3
=(α+β)4-4α3β-4αβ3-6α2β2
=(α+β)4-4αβ(α2+β2)-6(αβ)2
=(α+β)4-4αβ[(α+β)2-2αβ]-6(αβ)2
=(-b)4-4c[(-b)2-2c]-6(c)2
=b4-4c(b2-2c)-6c2
=b4-4b2c+8c2-6c2
=b4-4b2c+2c2

1)

Sum of root : - b

Product of root : c

a^4 + p^4

= (a^2 - ap + p^2)(a^2 + ap + p^2)

= [(a + p)^2 - 3ap][(a + p)^2 - ap]

= (b^2 - 3c)(b - c)