Question

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

Answer 1

Let  f (x) = x³ + 3x² − 2αx + β be the given polynomial.

By the factor theorem, (x+1) and  (x+2) are the factor of the polynomial f(x) if and  f (-2) (f(-1))both are equal to zero.

Therefore,

\[f( - 1) = ( - 1 )^3 + 3( - 1 )^2 - 2\alpha\left( - 1 \right) + \beta = 0\]

\[ \Rightarrow f( - 1) = - 1 + 3 + 2\alpha + \beta = 0\]

\[ \Rightarrow 2\alpha + \beta = - 2 . . . (i)\]

\[f( -2 ) = ( - 2 )^3 + 3( - 2 )^2 - 2\alpha\left( - 2 \right) + \beta = 0\]

                                     \[   - 8 + 12 + 4\alpha + \beta = 0\]

                                                         \[ 4\alpha + \beta = - 4 . . . (ii)\]

Subtracting (i) from (ii)

We get,

`(4alpha +beta) - (2alpha + beta) = -2`

                                         `2alpha = -2`

\[\alpha = - 1\]

Putting the value of \[\alpha\]

in equation (i), we get

`2 xx (-1) + beta = -2`

             ` - 2 + beta = -2`

                        `beta = 0`

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