Find α and β if x + 1 and x + 2 are factors of x3 + 3x2 - 2αx + �

1.	Find α and β if x + 1 and x + 2 are factors of x3 + 3x2 - 2αx + β.
Answer» 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.

Find α and β if x + 1 and x + 2 are factors of x3 + 3x2 - 2αx + β.

Answer»

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.