z=λ³+(i-4)λ² +2 -175i
z = λ³-4λ² + 2 + i (λ² - 175 )
Principle argument of z = tan-1( (λ² - 175 )/(λ³- 4λ² + 2) )
for Principle argument to be positive
Case 1: Numerator and Denominator Negative
(λ² - 175 )/(λ³-4λ² + 2) > 0
(λ² - 175 )/[λ²(λ-4) +2] > 0
(λ² - 175 ) is negative for λ [-13, 13]
[λ²(λ-4) + 2] is negative for λ [3, 1] U [-1, -infinity )
intersection of above two intervals, λ [-13, -1] U [1, 3] -- Eqn 1
Case 1: Numerator and Denominator Positive
(λ² - 175 ) is Positive for λ [14, infinity ) { total value of (λ² - 175 )/(λ³- 4λ² + 2) almost zero. }
[λ²(λ-4) + 2] is Positive for λ {0} U [4, infinity )
intersection of above two intervals, no common interval -- Eqn 2
from Eqn 1 and Eqn 2
number of λ values for which argument of is positive = 16
