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How to factor this polynomial?

M M
Posted 1 year ago

I want to factorize the following polynomial w.r.t variable x: 

 (x^2 \[Nu] \[Sigma]^2)/(
     2 (1 - \[CapitalTheta]))- (i x \[Mu] \[Nu])/(1 - \[CapitalTheta]) +1

I want to be the degree of each factored part 1, like (Cx-1)(Dx-1) where C and D are based on other variables. How can I do this in mathematica? thanks for yoru help in advance.
POSTED BY: M M
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I don't think you can get automatic steps for this kind of elementary calculations. I don't know a way to make Mathematica discover the factorization a x^2 + b x + c == a (x - x1) (x - x2) when the coefficients are symbolic.
POSTED BY: Gianluca Gorni
M M
M M
Posted 1 year ago

thank you so much, dear Gianluca. your way is very helpful. Is it possible for mathematica to show the steps of the solution as well for this problem? If yes, How can I see these steps?

thanks.
POSTED BY: M M

Here is a way:

pol = Collect[(x^2 \[Nu] \[Sigma]^2)/(2 (1 - \[CapitalTheta])) -
   (i x \[Mu] \[Nu])/(1 - \[CapitalTheta]) + 1, x]
firstCoeff = Coefficient[pol, x^2]
sols = SolveValues[pol == 0, x]
factorizedPol = Simplify[firstCoeff*sols[[1]] sols[[2]]]*
  (x/sols[[1]] - 1) (x/sols[[2]] - 1)
Simplify[factorizedPol == pol]
POSTED BY: Gianluca Gorni
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