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

Answer: \(( α+1)(β +1)=1-q\)

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 \(ax^2+bx+c\), we get

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

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

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

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

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

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

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

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 \(\mathbf{(\alpha+1)(\beta+1)}\) is \(\mathbf{1-q}\).