Question

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

Answer 1

Let be the given polynomial.

By the factor theorem, and are the factor of the polynomial f(x) if and both are equal to zero.

Therefore,

$$\begin{gathered} f(-1)=(-1{)}^3+3(-1{)}^2-2\alpha \left(-1\right)+\beta =0 \\ \Rightarrow f(-1)=-1+3+2\alpha +\beta =0 \\ \Rightarrow 2\alpha +\beta =-2...\left(\mathrm{i}\right) \end{gathered}$$

and

Subtracting (i) from (ii)

We get,

$\alpha =-1$

Putting the value of $\alpha$ in equation (i), we get

Hence, the value of α and β are −1, 0 respectively.