Question

If a and b are the zeroes of the polynomial f (X)=x² - 5x +k such that a-b =1 , find the value of k?

Answer 1

Since a,b,c usually represent the coeff of the terms of a polynomial, let us take α and β as the zeroes.

Given α and β are the zeroes of the polynomial x² − 5x + k
Also given that α − β = 1 → (1)
Recall that sum of roots (α + β) = −(b/a)
∴ α + β = 5 → (2)
Add (1) and (2), we get
α − β = 1
α + β = 5
2α = 6
∴ α = 3
Put α = 3 in α + β = 5
3 + β = 5
∴ β = 2
Hence 3 and 2 are zeroes of the given polynomial
Put x = 2 in the given polynomial to find the value of k ( Since 2 is a zero of the polynomial, f(2) will be 0 )

x² − 5x + k = 0
⇒ 2² − 5(2) + k = 0
⇒ 4 − 10 + k = 0
⇒ − 6 + k = 0
∴ k = 6

Hope this helps :)