If α, β are the zeroes of the polynomial $f(x)=x^2-p(x+1)-q$, then ( α+1)(β +1)=

Answer: $(\alpha +1)(\beta +1)=1-q$

Solution:

Given: α, β are the zeroes of the polynomial $f(x)=x^2-p(x+1)-q$.

Step 1: Use the relationship between coefficients and roots of quadratic polynomial.

$f(x)=x^2-p(x+1)-q$

$\Rightarrow f(x)=x^2-px-p-q$

$\Rightarrow f(x)=x^2-px-(p+q)$

By comparing the polynomial $f(x)=x^2-px-(p+q)$ with standard quadratic polynomial $a{x}^2+bx+c$, we get

$a=1,b=-p$ and $c=-(p+q)$.

Sum of roots $(\alpha +\beta )=-\frac{b}{a}=$

$\Rightarrow \alpha +\beta =-\frac{-p}{1}=p$

Product of roots $(\alpha \cdot \beta )=\frac{c}{a}$

$\Rightarrow \alpha \cdot \beta =\frac{-(p+q)}{1}=-p-q$

Step 2: Expand the expression $\mathbf{(}\alpha \mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\beta \mathbf{+}\mathbf{1}\mathbf{)}$.

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

Substitute the value of $\alpha \cdot \beta$ and (\alpha+\beta\), we get

$(\alpha +1)(\beta +1)=p+(-p-q)+1$

$\Rightarrow (\alpha +1)(\beta +1)=p-p-q+1$

$\Rightarrow (\alpha +1)(\beta +1)=-q+1$

$\Rightarrow (\alpha +1)(\beta +1)=1-q$

Hence, the value of $\mathbf{(}\alpha \mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\beta \mathbf{+}\mathbf{1}\mathbf{)}$ is $\mathbf{1}\mathbf{-}\mathbf{q}$.