Finding the Value of c Using Zeroes of a Quadratic Polynomial

Video Explanation

Question

If \( \alpha \) and \( \beta \) are the zeroes of the polynomial

\[ f(x) = x^2 – p(x+1) – c, \]

such that

\[ (\alpha + 1)(\beta + 1) = 0, \]

find the value of \(c\).

Options:

(a) 1    (b) 0    (c) -1    (d) 2

Solution

Step 1: Write the Polynomial in Standard Form

\[ f(x) = x^2 – px – p – c \]

Step 2: Use Relations Between Zeroes and Coefficients

\[ \alpha + \beta = p, \quad \alpha\beta = -(p + c) \]

Step 3: Evaluate the Given Expression

\[ (\alpha + 1)(\beta + 1) = \alpha\beta + (\alpha + \beta) + 1 \]

\[ = [-(p + c)] + p + 1 = 1 – c \]

Step 4: Use the Given Condition

\[ 1 – c = 0 \Rightarrow c = 1 \]

Conclusion

The correct value of \(c\) is:

\[ \boxed{1} \]

Hence, the correct option is (a) 1.