The equation x^3-x^2+ax+b=0 has three integer solutions. Two of these solutions are x=1 and x=2. Work out the third solution to the equation.

Question

The equation
`x^3-x^2+ax+b=0` has three integer solutions.

Two of these solutions are
`x=1` and
`x=2`.

Work out the third solution to the equation.

Answer 1

x = - 2

Explanation:

x³-x²+ax+b = (x-1)(x-2)(x+c) = x³ + (c-3)x² + (2-3c)x +2c

c-3 = -1 C = 2

a = 2 - 3c = 2 - 3*2 = - 4

b = 2c = 4

third solution: x+2=0 x=-2