If α and β are the zeros of the quadratic polynomial f(x) = x² − p(x + 1) − c, show that (α + 1)(β + 1) = 1 − c

Video Explanation

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Proof

Given polynomial:

f(x) = x² − p(x + 1) − c

Simplifying the polynomial:

f(x) = x² − px − p − c

Let α and β be the zeros of the given polynomial.

Step 1: Find α + β and αβ

Comparing f(x) = x² − px − (p + c) with ax² + bx + c:

a = 1,   b = −p,   c = −(p + c)

α + β = −b/a = p

αβ = c/a = −(p + c)

Step 2: Find (α + 1)(β + 1)

(α + 1)(β + 1)

= αβ + α + β + 1

= −(p + c) + p + 1

= 1 − c

Hence Proved

(α + 1)(β + 1) = 1 − c

Conclusion

Thus, using the relationship between zeros and coefficients of the quadratic polynomial, the given result is proved.