zeroes of polynomial x cube - x square - x + 2
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P(x) = x³ - x² - x + 2
let y = x - 1/3 So x = y + 1/3
P(y) = y³ + y² + y/3 + 1/27 - y² - 1/9 - 2y/3 - y - 1/3 + 2
P(y) = y³ - 3p y + q = 0 , where p = 4/9 and q = 43/27
Let y = s + p/s = (s² + p)/s
s³ * P(s) = (s⁶ + 3 s⁴p + 3 s² p² + p³) - (3 p s⁴ + 3 p² s²) + s³ q = 0
=> s⁶ + s³ q + p³ = 0
s³ = [ - 43/27 +- √(43²/27^2 - 4*4³/9³) ] / 2
s = -1.155278 or, - 0.37007
x = y + 1/3 = s + p/s +1/3 = -1.205556
P(x) = (x+1.205556) (x² - 2.205556 x + 2/1.205556)
The other roots are : imaginary as discriminant < 0.
roots: one negative. -1.205556
let y = x - 1/3 So x = y + 1/3
P(y) = y³ + y² + y/3 + 1/27 - y² - 1/9 - 2y/3 - y - 1/3 + 2
P(y) = y³ - 3p y + q = 0 , where p = 4/9 and q = 43/27
Let y = s + p/s = (s² + p)/s
s³ * P(s) = (s⁶ + 3 s⁴p + 3 s² p² + p³) - (3 p s⁴ + 3 p² s²) + s³ q = 0
=> s⁶ + s³ q + p³ = 0
s³ = [ - 43/27 +- √(43²/27^2 - 4*4³/9³) ] / 2
s = -1.155278 or, - 0.37007
x = y + 1/3 = s + p/s +1/3 = -1.205556
P(x) = (x+1.205556) (x² - 2.205556 x + 2/1.205556)
The other roots are : imaginary as discriminant < 0.
roots: one negative. -1.205556
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