Evaluation Using Zeros of a Quadratic Polynomial

Video Explanation

Question

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

\[ f(x) = x^2 – p(x+1) – c, \]

find

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

Solution

Step 1: Write the Polynomial in Standard Form

\[ f(x) = x^2 – px – p – c \]

Step 2: Use Relations Between Zeroes and Coefficients

For a quadratic polynomial \(ax^2 + bx + d\),

\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{d}{a} \]

Here,

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

Step 3: Evaluate the Required Expression

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

\[ = \big[-(p + c)\big] + p + 1 \]

\[ = 1 – c \]

Conclusion

\[ \boxed{(\alpha + 1)(\beta + 1) = 1 – c} \]