Find α, β if (x + 1) and (x + 2) are the factors

Solution:

Given, $f(x)=x^{3}+3 x^{2}-2 \alpha x+\beta$ and the factors are $(x+1)$ and $(x+2)$

From factor theorem, if they are the factors of f(x) then results of f(-2) and f(-1) should be zero

Let, x + 1 = 0

⟹ x = -1

Substitute value of x in f(x)

$f(-1)=(-1)^{3}+3(-1)^{2}-2 \alpha(-1)+\beta$

= −1 + 3 + 2α + β

= 2α + β + 2 ... 1

Let, x + 2 = 0

⟹ x = -2

Substitute value of x in f(x)

$f(-2)=(-2)^{3}+3(-2)^{2}-2 \alpha(-2)+\beta$

= −8 + 12 + 4α + β

= 4α + β + 4 .... 2

Solving 1 and 2 i.e (1 - 2)

⟹ 2α + β + 2 - (4α + β + 4) = 0

⟹ −2α - 2 = 0

⟹ 2α = −2

⟹ α = −1

Substitute α= -1 in equation 1

⟹ 2(−1) + β = -2

⟹ β = -2 + 2

⟹ β = 0

The values are α = −1 and β = 0