f-x-x-4-2x-3-3x-2-4x-5-Find-the-roots-

Question Number 39779 by ajfour last updated on 10/Jul/18

f (x) = x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 5

F i n d t h e r o o t s .

Answered by MJS last updated on 11/Jul/18

approximate :

x_{1} = - . 287815 - 1 . 41609 i

x_{2} = - . 287815 + 1 . 41609 i

x_{3} = 1.28782 - . 857897 i

x_{4} = 1.28782 + . 857897 i

exact :

the usual procedure, but we have two pairs

of conjugated complex solutions

f (x) = (x - α - \sqrt{β} i) (x - α + \sqrt{β} i) (x - γ - \sqrt{δ} i) (x - γ + \sqrt{δ} i)

\Rightarrow α = 1 - γ

β = \frac{γ (2 γ^{2} - 3 γ + 3)}{2 γ - 1}

δ = \frac{(γ - 1) (2 γ^{2} - γ + 2)}{2 γ - 1}

γ^{6} - 3 γ^{5} + \frac{9}{2} γ^{4} - 4 γ^{3} + \frac{21}{16} γ^{2} + \frac{3}{16} γ - \frac{3}{16} = 0

γ = u + \frac{1}{2}

u^{6} + \frac{3}{4} u^{4} - \frac{3}{4} u^{2} - \frac{1}{16} = 0

u = \sqrt{v}

v^{3} + \frac{3}{4} v^{2} - \frac{3}{4} v - \frac{1}{16} = 0

v = w - \frac{1}{4}

w^{3} - \frac{15}{16} w + \frac{5}{32} = 0

this has got 3 real solutions, so we need the

trigonometric method .

w_{k} = 2 \sqrt{- \frac{p}{3}} \sin (\frac{1}{3} (\arcsin (\frac{9 q}{2 p^{2}} \sqrt{- \frac{p}{3}}) + 2 k π)) with k = 0, 1, 2

p = - \frac{15}{16}; q = \frac{5}{32}

w_{0} = \frac{\sqrt{5}}{2} \sin (\frac{1}{3} \arcsin \frac{\sqrt{5}}{5})

w_{1} = \frac{\sqrt{5}}{2} \cos (\frac{π}{6} + \frac{1}{3} \arcsin \frac{\sqrt{5}}{5})

w_{2} = - \frac{\sqrt{5}}{2} \sin (\frac{π}{3} + \frac{1}{3} \arcsin \frac{\sqrt{5}}{5})

again the way back is hard if you want

exact solutions . all three w_{k} lead to the

same set of α, β, γ, δ but I recommend to

take w_{1} to get real values in each step back .

Commented by MJS last updated on 10/Jul/18

Ferrari is very complicated sometimes but

yes, it always gives all solutions . Still the

problem witb the 3 real solutions occurs \dots

Commented by ajfour last updated on 10/Jul/18

I r e a d a l i t t l e o f {F e r r a r i}^{'} s s o l u t i o n .

Commented by ajfour last updated on 11/Jul/18

T h a n k s f o r t h i s m u c h, S i r .

W h a t s t h e t r i g o n o m e t r i c w a y

f o r o u r c u b i c e q . ?

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