+128475 Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = x^2 - 5x + 3. Compute a^3*b + a*b^3.
Note that
a^3b + ab^3 = ab ( a^2 + b^2)
Rearrange equation as x^2 - 3x + 4 = 0
By Vieta
Sum of roots = -(-3) /1 = 3 and product of roots = 4/1 = 4
So
a + b = 3 square both sides and ab = 4 so 2ab = 8
a^2 + 2ab + b^2 = 9
a^2 + b^2 = 9 - 2ab
a^2 + b^2 = 9 - 2(8) = -7
So
a^3b + ab^3 = ab ( a^2 + b^2) = 4 ( -7) = -28
