If α, β are the roots of the quadratic equation x2 + px + q = 0, then the values of α3, β3 and α4+α2β3+β4 are respectively

Question

If α, β are the roots of the quadratic equation x2 + px + q = 0, then the values of α3, β3 and α4+α2β3+β4 are respectively (A) 3pq - p3 and p4 - 3p2q + 3q2 (B) -p(3q - p2) and (p2 - q)(p2 + 3q) (C) pq - 4 and p4 - q4 (D) 3pq - p3 and (p2 - q) (p2 - 3q)

Tardigrade Admin · Accepted Answer

∵ Sum of roots, α+β=-p and α β=q ∴ (α3+β3)=(α+β)3-3 α β(α+β) =(-p)3-3 q(-p) =-p3+3 p q and α4+α2 β2+β4=(α4+β4)+(α β)2 =(α2+β2)2-(α β)2 = [(α+β)2-2 α β]2-(α β)2 =[(-p)2-2 q]2-3(q)2 =[p2-2 q]2-3 q2 =p4-4 p2 q+4 q2-q2 =p4-4 p2 q+3 q2 =p4-3 p2 q-p2 q+3 q2 =p2(p2-3 q)-q(p2-3 q) = (p2-q)(p2-3 q)

Q. If $α, β$ are the roots of the quadratic equation $x^{2} + p x + q = 0$ , then the values of $α^{3}, β^{3}$ and $α^{4} + α^{2} β^{3} + β^{4}$ are respectively

$3 pq - p^{3}$ and $p^{4} - 3 p^{2} q + 3 q^{2}$

$- p (3 q - p^{2})$ and $(p^{2} - q) (p^{2} + 3 q)$

$pq - 4$ and $p^{4} - q^{4}$

$3 pq - p^{3}$ and $(p^{2} - q) (p^{2} - 3 q)$

Q. If α,β are the roots of the quadratic equation x2+px+q=0, then the values of α3,β3 and α4+α2β3+β4 are respectively

3pq−p3 and p4−3p2q+3q2

−p(3q−p2) and (p2−q)(p2+3q)

pq−4 and p4−q4

3pq−p3 and (p2−q)(p2−3q)

Q. If $α, β$ are the roots of the quadratic equation $x^{2} + p x + q = 0$ , then the values of $α^{3}, β^{3}$ and $α^{4} + α^{2} β^{3} + β^{4}$ are respectively

$3 pq - p^{3}$ and $p^{4} - 3 p^{2} q + 3 q^{2}$

$- p (3 q - p^{2})$ and $(p^{2} - q) (p^{2} + 3 q)$

$pq - 4$ and $p^{4} - q^{4}$

$3 pq - p^{3}$ and $(p^{2} - q) (p^{2} - 3 q)$