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

Video Explanation

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Given

f(x) = x² − p(x + 1) − c

α and β are the zeros of the polynomial.

To Find

The value of (α + 1)(β + 1).

Solution

Step 1: Simplify the Given Polynomial

f(x) = x² − p(x + 1) − c

= x² − px − p − c

Comparing with the standard form ax² + bx + c:

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

Step 2: Find α + β and αβ

For a quadratic polynomial:

α + β = −b/a

αβ = constant term / a

∴ α + β = −(−p)/1 = p

αβ = −(p + c)

Step 3: Find (α + 1)(β + 1)

(α + 1)(β + 1) = αβ + α + β + 1

= [−(p + c)] + p + 1

= 1 − c

Final Answer

(α + 1)(β + 1) = 1 − c

Conclusion

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