Question

If a and b are the zeroes of the polynomial $f\left(x\right)=x^2-p(x+1)-c$, then $(\alpha +1)(\beta +1)$ is equal to _______.

• A. c - 1
• B. 1 - c
• C. c
• D. 1 + c

Answer 1

Answer: (B)

1 - c

The given polynomial can be rearranged as follows,

$f\left(x\right)=x^2-px-(p+c)$

The zeroes of the polynomial are $\alpha$ and $\beta$

So, the sum of the roots is $p$ and product of the roots is $-(p+c)$

We need to find $(\alpha +1)(\beta +1)$

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

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

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

Therefore,Option B is correct.