Question

If $\alpha$ and $\beta$ are zeroes of the polynomial $x^2-a(x+1)-b$ such that $(\alpha +1)(\beta +1)=0,$ find the value of $b.$

Answer 1

Given polynomial is

$x^2-a(x+1)-b=0$

$x^2-ax-a-b=0$

$x^2-ax-(a+b)=0$

$\alpha +\beta =a,\alpha \beta =-a-b$.......... given

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

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

$=-a-b+a+1=0\Rightarrow b=1$