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?

- anonymous

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- Loser66

What is Descartes' rule?

- 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

http://www.algebra.com/algebra/homework/complex/Complex_Numbers.faq.question.764641.html

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## More answers

- Loser66

You don't understand it, right?

- anonymous

I don't.

- Loser66

let's look at example:
\(x^5-x^4+x+2=0\) how many time the sign of the term change?

- Loser66

ok!!
what is the sign of x^5?

- Loser66

+ or - ??

- anonymous

positive

- Loser66

what is the sign of x^4 ?

- anonymous

- lol

- Loser66

yes, so, if you go from x^5 to x^4 the sign changes from + to -, right? one time!!

- Loser66

now, next , what is the sign of x ?

- anonymous

Got it! Basically the number of signs changes is the number of possible roots, correct?

- anonymous

+

- Loser66

possible of REAL ROOT

- Loser66

OK, tell me, on the expression above, how many time the sign change in total?

- anonymous

2

- Loser66

yup, so the POSSIBLE real roots are ???

- anonymous

2 :)

- Loser66

or 0

- anonymous

So this is the Descartes rule?

- Loser66

YUp

- anonymous

How come it's either 0?

- 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

multiple of 2, right?

- Loser66

yup

- Loser66

and the complex root:

- Loser66

What is the degree of the expression above?

- anonymous

5

- Loser66

yup, so the number of real root are 2 (maximum), hence the maximum of complex is ??

- anonymous

Are we now talking about the Fundamental Theorem of Algebra?

- Loser66

yup

- anonymous

Okay. :) So in the Fundamental Theorem of Algebra, it's about the complex numbers?

- Loser66

complex roots, yes

- anonymous

roots, I meant. :)

- Loser66

ok, how many??

- anonymous

Okay, so the equation above has a degree of 5.

- Loser66

yup

- anonymous

i already answered that lol

- Loser66

degree 5--> maximum 5 roots, we already know that it MAY have 2 real roots, hence, how many complex left?

- anonymous

and 2 complex roots, since the real roots are 2

- Loser66

perfect!! but confirm: why 2 but 3??

- anonymous

because complex roots are always in pairs?

- anonymous

idk haha

- Loser66

yyyyyyyyyyyyyyyyyyyyyyyyyyyyyes!

- Loser66

You got it.

- anonymous

oh really?

- anonymous

ok so what now?

- Loser66

go to bed!! we are done.

- anonymous

That's it?

- Loser66

yes, dat sit.

- anonymous

wow, i didn't know that that's easy. thank you @Loser66 ! :) You're a great help!

- Loser66

np

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