+452 Let p, q, r, and s be the roots of g(x) = 3x^4 - 8x^3 + 5x^2 + 2x - 17 - 2x^4 + 10x^3 + 11x^2 + 18x - 14.
Compute p^2 + q^2 + r^2 + s^2.
+130462 Simplify as
x^4 + 2x^3 + 16x^2 + 20x - 31
By Viete :
p + q + r + s = -2
(pr + pq + ps + qr + qs + rs) = 16
So
( p + q + r + s)^2 = (-2)^2 = 4 =
(p^2 + q^2 + r^2 + s^2) + 2(pr + pq + ps + qr + qs + rs)
And
2 (pr + pq + ps + qr + qs + rs) = 2(16) = 32
So
( p + q + r + s)^2 =
(p^2 + q^2 + r^2 + s^2) + 2(pr + pq + ps + qr + qs + rs) =
(p^2 + q^2 + r^2 + s^2) + 32 = 4
(p^2 + q^2 + r^2 + s^2) = -28
