Let, f (x) = x3 + 3x2 - 2αx + β be the given polynomial,
From factor theorem,
If (x + 1) and (x + 2) are factors of f (x) then f (-1) = 0 and f (-2) = 0
f (-1) = 0
(-1)3 + 3 (-1)2 – 2 α (-1) + β = 0
-1 + 3 + 2 α + β = 0
2 α + β + 2 = 0 (i)
Similarly,
f (-2) = 0
(-2)3 + 3 (-2)2 – 2 α (-2) + β = 0
-8 + 12 + 4 α + β = 0
4 α + β + 4 = 0 (ii)
Subtract (i) from (ii), we get
4 α + β + 4 – (2 α + β + 2) = 0 – 0
4 α + β + 4 - 2 α - β - 2 = 0
2 α + 2 = 0
α = -1
Put α = -1 in (i), we get
2 (-1) + β + 2 = 0
β = 0
Hence,
α = -1 and β = 0.