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 (x−i) 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 (x−i)(x+i)=x2+1 you will get x3+8 which factors as (x3+8)=(x+2)(x2−2x+4) which has a solution of x=−2
Now use quadratic formula to solve x2−2x+4=0 to find other roots and factor if you wish.
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