If α and β are the zeros of the polynomial f(x) = x² - 5x + k such that α - β = 1, find the value of k.

Question: If α and β are the zeros of the polynomial f(x) = x² - 5x + k such that α - β = 1, find the value of k.

Solution:
Given,
f(x) = x² - 5x + k
α - β = 1 .................... (1)
α and β are the zeros of the polynomial.
α + β = - (- 5)
∴ α + β = 5......................... (2)

(1) + (2)
α - β = 1
α + β = 5
2α = 6
∴ α = 3

From equ. (2)
β = 5 - 3
∴ β = 2

Now, αβ = k
⇒ k = 3 × 2
∴ k = 6