(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Course - Control systems engineering, chapter: design using root locus

I'm familiar with dividing a polynomial when given a factor using the remainder theorem however is there another way when only the third or fourth order equation is given and nothing else? We aren't allowed to use calculators in our exams.

Any help?

2. Relevant equations

none - rules of basic factorization, remainder theorem, if theres another than I'd be happy to learn of it.

3. The attempt at a solution

My method is quick for simple non complex numbers. However since its for a control systems exam i'm worried that they will definitely throw in a complex root set thats gonna make me wish i had my TI - 8# calculator in the exam.

my method is simply :

say we have

(s+1)(s+2)(s+3)

= (s^2 + 3s + 2 )(s+3)

= s^3 + 3s^2 + 2s + 3s^2 + 9s + 6

= s^3 + 6s^2 + 11s + 6

so now that we know what our answer should be - lets start

1) look at the final number of the polynomial without s terms - in this case its 6

we find theGCDfor it - being

3 |_6_

......2

2) so ... lets substitute -3 into the above equation and see if we obtain 0.

s^3 + 6s^2 + 11s + 6 | s = -3

= -27 + 54 -33 + 6 = -60 + 60 = 0

so therefore -3 is a root so we now have

(s+3) as one of the factors

3) rewrite the original polynomial in terms of (s+3)

s^3 + 6s^2 + 11s + 6

= s^2(s+3) + 3s(s+3) + 2(s+3)

= (s+3)(s2 + 3s +2)

= (s+3)(s+1)(s+2)<--- as its a 2nd order simple polynomial(quadratic) we can factorize by inspection

done.

But as you can see i need to first check if its a zero then split up the terms - its very traditional and I've used it since high school however I'm sure it won't work for complex roots.

can someone find a faster method? Hopefully one that incorporates complex factors?

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# Homework Help: Whats a faster way for factorizing polynomials of order 3 and above

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