If α and β are the zeros of the polynomial f(x) = x² − p(x + 1) − c such that (α + 1)(β + 1) = 0, find the value of c

Video Explanation

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Given

f(x) = x² − p(x + 1) − c

α and β are the zeros of the polynomial.

(α + 1)(β + 1) = 0

To Find

The value of c.

Solution

Step 1: Simplify the Polynomial

f(x) = x² − p(x + 1) − c

= x² − px − p − c

Step 2: Find α + β and αβ

Comparing with ax² + bx + d:

a = 1,   b = −p,   constant term = −(p + c)

Sum of zeros:

α + β = −b/a = p

Product of zeros:

αβ = −(p + c)

Step 3: Use the Given Condition

(α + 1)(β + 1) = αβ + α + β + 1

= [−(p + c)] + p + 1

= 1 − c

Given that:

(α + 1)(β + 1) = 0

∴ 1 − c = 0

∴ c = 1

Final Answer

The value of c is:

c = 1

Correct Option

(a) 1

Conclusion

Hence, if (α + 1)(β + 1) = 0 for the polynomial f(x) = x² − p(x + 1) − c, then the value of c is 1.