If zeroes of cubic polynomial f(x)=kx^3-8x^2+5 …
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Gaurav Seth 6 years, 7 months ago
k = 8/3α
α = √15 / 4
Step-by-step explanation:
f(x) = kx³ - 8x² + 5
Roots are α - β , α & α +β
Sum of roots = - (-8)/k
Sum of roots = α - β + α + α +β = 3α
=> 3α = 8/k
=> k = 8/3α
or we can solve as below
f(x) = (x - (α - β)(x - α)(x - (α +β))
= (x - α)(x² - x(α+β + α - β) + (α² - β²))
= (x - α)(x² - 2xα + (α² - β²))
= x³ - 2x²α + x(α² - β²) - αx² +2α²x - α³ + αβ²
= x³ - 3αx² + x(3α² - β²) + αβ² - α³
= kx³ - 3αkx² + xk(3α² - β²) + k(αβ² - α³)
comparing with
kx³ - 8x² + 5
k(3α² - β²) = 0 => 3α² = β²
k(αβ² - α³) = 5
=>k(3α³ - α³) = 5
=> k2α³ = 5
3αk = 8 => k = 8/3α
(8/3α)2α³ = 5
=> α² = 15/16
=> α = √15 / 4
0Thank You