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Common Roots

Find α and ...

Question

Find $α$ and $β$ , if $x + 1$ and $x + 2$ are the factors of $x^{3} + 3 x^{2} - 2 α x + β$

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Solution

if we put $x = - 1, x = - 2$ then the equation will be equal to zero
put $x = - 1$
$- 1 + 3 + 2 α + β = 0$
$2 α + β = - 2$
now put $x = - 2$
$- 8 + 12 + 4 α + β = 0$
$- 4 α - β = 4$
add the 2 equations
$- 2 α = 2$
$α = - 1, β = 0$

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