Given equation is
x3 – 2x2 – 5x + 6 = 0
since, 1 is root of the equation, therefore (x + 1) is a factor of it.
⇒ (x – 1)(x2 – x – 6) = 0
⇒ (x – 1)(x2 – 3x + 2x – 6) = 0
⇒ (x – 1)(x(x – 3) + 2(x – 3)) = 0
⇒ (x – 1)(x + 2)(x – 3) = 0
⇒ x = 1, –2 or 3.
Hence, 1 –2 & 3 are roots of given equation.
Then α & β are –2 & 3.
i.e, either α = –2 & β = 3
or α = 3 & β = –2.