Question

give the solution of this question​

Answer 1

$\: \: \: \: \: {x}^{2} - a(x + 1) - b \\ = {x}^{2} - ax - a - b \\ = {x}^{2} + ( - a)x + ( - a - b)$

On comparing with the general form i.e.

Ax² + Bx + C, we get

A = 1

B = -a

C = - a - b

$since \: \alpha \: and \: \beta \: are \: the \: roots \: of \: the \: given \\ polynomial \\ \\ \alpha + \beta = \frac{ - b}{a} = \frac{ - ( - a)}{1} = a \\ \alpha \beta = \frac{c}{a} = \frac{ - a - b}{1} = - a - b \\ \\ now \\ \: \: \: \: \: ( \alpha + 1)( \beta + 1) \\ = \alpha \beta + \alpha + \beta + 1 \\ = - a - b + a + 1 \\ = - a + a + 1 - b \\ = 1 - b$

Hence,

$( \alpha + 1)( \beta + 1) = 1 - b$