If a and B are the zeroes of the polynomial p(x) = 6x² + 5x - k satisfying the relation, a - B = 1, then find the value of k.
Answers
Answer:
If α and β are the zeros of the equation,
It means that :
x = α or x = β
Since α - β = 1, therefore
α = β + 1
x= β + 1 or x = β
The factors of x2 - 5x + k are (x - β) and (x - β - 1)
(x - β)*(x - β - 1) = x2 - 5x + k
x2 - βx - x - βx + β2 + β = x2 - 5x + k
x2 - 2βx - x + β2 + β = x2 - 5x + k
Subtract x2 from both sides,
- 2βx - x + β2 + β = - 5x + k
Equating the like terms on both sides, we see that
- 2βx - x = - 5x (equation 1)and
β2 + β = k (equation 2)
From equation 1,
- 2βx - x = - 5x
- x(2β + 1) = - 5x
Divide both sides by -x
2β + 1 = 5
2β = 4, therefore β = 2
Putting β as 2 in equation 2, we have
2^2 + 2 = k
k = 6
To confirm, we first find α,
α = β + 1
α = 2 + 1 = 3
(x -α) (x - β) = x2 - 5x + k
(x -3) (x - 2) = x2 - 5x + k
x2 - 2x - 3x + 6 = x2 - 5x + k
x2 - 5x + 6 = x2 - 5x + k
Therefore, k = 6
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