Question

If α and β are the roots of x²-px+q, find the values of
i) α²+β²
ii) α³+β³

Answer 1

If α,β are the roots of the polynomial x2 - px + q.

α + β = p and αβ = q

if the roots are (α2 -β2)(α3 - β3) and (α3β2 + α2β3)

= (α2 -β2)(α3 - β3) + (α3β2 + α2β3)

= (α +β)(α -β)2(α2 + αβ+ β2) + α2β2(α + β).

= (α +β)[ (α +β)2 - 4αβ ] [ (α +β)2 - αβ ] + α2β2(α + β).

= p( p2 - 4q)( p2 - q) + q2p

= p( p4 - p2q -4p2q + 4q2) + pq2

= p5 - 5p3q + 5pq2 .

⇒ (α2 -β2)(α3 - β3)(α3β2 + α2β3) = (p6q2-5p4q3+4p2q4)

∴ x2- (p5 - 5p3q + 5pq2 ) x + ( p6q2 - 5p4q3 + 4p2q4 ) = 0.

Answer 2

please see the image to get answer