anonymous
  • anonymous
How could you use Descartes' rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial as well as find the number of possible positive and negative real roots to a polynomial?
Mathematics
schrodinger
  • schrodinger
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Loser66
  • Loser66
What is Descartes' rule?
anonymous
  • anonymous
possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients of the terms or less than the sign changes by a multiple of 2.
LynFran
  • LynFran
http://www.algebra.com/algebra/homework/complex/Complex_Numbers.faq.question.764641.html

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Loser66
  • Loser66
You don't understand it, right?
anonymous
  • anonymous
I don't.
Loser66
  • Loser66
let's look at example: \(x^5-x^4+x+2=0\) how many time the sign of the term change?
Loser66
  • Loser66
ok!! what is the sign of x^5?
Loser66
  • Loser66
+ or - ??
anonymous
  • anonymous
positive
Loser66
  • Loser66
what is the sign of x^4 ?
anonymous
  • anonymous
- lol
Loser66
  • Loser66
yes, so, if you go from x^5 to x^4 the sign changes from + to -, right? one time!!
Loser66
  • Loser66
now, next , what is the sign of x ?
anonymous
  • anonymous
Got it! Basically the number of signs changes is the number of possible roots, correct?
anonymous
  • anonymous
+
Loser66
  • Loser66
possible of REAL ROOT
Loser66
  • Loser66
OK, tell me, on the expression above, how many time the sign change in total?
anonymous
  • anonymous
2
Loser66
  • Loser66
yup, so the POSSIBLE real roots are ???
anonymous
  • anonymous
2 :)
Loser66
  • Loser66
or 0
anonymous
  • anonymous
So this is the Descartes rule?
Loser66
  • Loser66
YUp
anonymous
  • anonymous
How come it's either 0?
Loser66
  • Loser66
that is the rule, if the number of the changing of the sign is 6, then the number of real root can be 6,4,2,0
anonymous
  • anonymous
multiple of 2, right?
Loser66
  • Loser66
yup
Loser66
  • Loser66
and the complex root:
Loser66
  • Loser66
What is the degree of the expression above?
anonymous
  • anonymous
5
Loser66
  • Loser66
yup, so the number of real root are 2 (maximum), hence the maximum of complex is ??
anonymous
  • anonymous
Are we now talking about the Fundamental Theorem of Algebra?
Loser66
  • Loser66
yup
anonymous
  • anonymous
Okay. :) So in the Fundamental Theorem of Algebra, it's about the complex numbers?
Loser66
  • Loser66
complex roots, yes
anonymous
  • anonymous
roots, I meant. :)
Loser66
  • Loser66
ok, how many??
anonymous
  • anonymous
Okay, so the equation above has a degree of 5.
Loser66
  • Loser66
yup
anonymous
  • anonymous
i already answered that lol
Loser66
  • Loser66
degree 5--> maximum 5 roots, we already know that it MAY have 2 real roots, hence, how many complex left?
anonymous
  • anonymous
and 2 complex roots, since the real roots are 2
Loser66
  • Loser66
perfect!! but confirm: why 2 but 3??
anonymous
  • anonymous
because complex roots are always in pairs?
anonymous
  • anonymous
idk haha
Loser66
  • Loser66
yyyyyyyyyyyyyyyyyyyyyyyyyyyyyes!
Loser66
  • Loser66
You got it.
anonymous
  • anonymous
oh really?
anonymous
  • anonymous
ok so what now?
Loser66
  • Loser66
go to bed!! we are done.
anonymous
  • anonymous
That's it?
Loser66
  • Loser66
yes, dat sit.
anonymous
  • anonymous
wow, i didn't know that that's easy. thank you @Loser66 ! :) You're a great help!
Loser66
  • Loser66
np

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