Question

The number of integers λ for which the equation x³ - 3x + λ = 0 has integer roots is

• A. 1
• B. 2
• C. 7
• D. Infinite

Answer 1

The correct option is B

2

Equation is $x^3-3x+\lambda =0.$
Let $\alpha ,\beta ,\gamma$ be the roots,then
$\alpha +\beta +\gamma =0\cdots \left(i\right)$
$\alpha \beta +\beta \gamma +\gamma \alpha =-3\cdots \left(ii\right)$
$\alpha \beta \gamma =-\lambda \cdots \left(iii\right)$
${\alpha }^2+{\beta }^2+{\gamma }^2=(\alpha +\beta +\gamma {)}^2-2(\alpha \beta +\beta \gamma +\gamma \alpha )$
Since $\alpha ,\beta ,\gamma$ are integers with $\alpha +\beta +\gamma =0$ and ${\alpha }^2+{\beta }^2+{\gamma }^2=6,$ we get,

$\left\{\begin{array}{ccc}\Rightarrow \alpha =2& \beta =-1& \gamma =-1\\ \alpha =-2& \beta =1& \gamma =1\end{array}\right\}\lambda =2\text{or}\lambda =-2$

$\lambda =2$ or $\lambda =-2.$