Wednesday, 17 April 2019

Finding remaining polynomial after finding complex factors



I want to express this polynomial as a product of linear factors:




x5+x3+8x2+8



I noticed that ±i were roots just looking at it, so two factors must be (xi) and (x+i), but I'm not sure how I would know what the remaining polynomial would be. For real roots, I would usually just do use long division but it turns out a little messy in this instance (for me at least) and was wondering if there was a simpler method of finding the remaining polynomial.



Apologies for the basic question!


Answer



If you divide x5+x3+8x2+8 by (xi)(x+i)=x2+1 you will get x3+8 which factors as (x3+8)=(x+2)(x22x+4) which has a solution of x=2



Now use quadratic formula to solve x22x+4=0 to find other roots and factor if you wish.



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