Q3/016) Given that kx^3+2x^2+2x+3 and kx^2-2x+9 have a common factor, what are the?

GCD( kx^3+2x^2+2x+3,kx^2-2x+9) is not 1

GCD( kx^3+2x^2+2x+3,kx^2-2x+9)
= GCD(kx^2 - 2x + 9, kx^3+2x^2+2x+3 - x(kx^2-2x+9))
= GCD((kx^2 - 2x + 9, 4x^2 - 7x + 3)

say f(x) = kx^2 -2x + 9

4x^2 - 7x + 3 = (x-1)(4x-3)

x =1 => f(x) = k - 2 + 9 = 0 or k = -7

If (4x - 3) is a common factor, then f(3/4)= 0.

f(3/4) = 0
=>9k/16 + 15/2 = 0
=>k = -40/3

So  k = -40/3 or - 7