Question

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?

Answer 1

What is Descartes' rule?

Answer 2

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.

Answer 3

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

Answer 4

You don't understand it, right?

Answer 5

I don't.

Answer 6

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

Answer 7

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

Answer 8

+ or - ??

Answer 9

positive

Answer 10

what is the sign of x^4 ?

Answer 11

- lol

Answer 12

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

Answer 13

now, next , what is the sign of x ?

Answer 14

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

Answer 15

+

Answer 16

possible of REAL ROOT

Answer 17

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

Answer 18

2

Answer 19

yup, so the POSSIBLE real roots are ???

Answer 20

2 :)

Answer 21

or 0

Answer 22

So this is the Descartes rule?

Answer 23

YUp

Answer 24

How come it's either 0?

Answer 25

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

Answer 26

multiple of 2, right?

Answer 27

yup

Answer 28

and the complex root:

Answer 29

What is the degree of the expression above?

Answer 30

5

Answer 31

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

Answer 32

Are we now talking about the Fundamental Theorem of Algebra?

Answer 33

yup

Answer 34

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

Answer 35

complex roots, yes

Answer 36

roots, I meant. :)

Answer 37

ok, how many??

Answer 38

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

Answer 39

yup

Answer 40

i already answered that lol

Answer 41

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

Answer 42

and 2 complex roots, since the real roots are 2

Answer 43

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

Answer 44

because complex roots are always in pairs?

Answer 45

idk haha

Answer 46

yyyyyyyyyyyyyyyyyyyyyyyyyyyyyes!

Answer 47

You got it.

Answer 48

oh really?

Answer 49

ok so what now?

Answer 50

go to bed!! we are done.

Answer 51

That's it?

Answer 52

yes, dat sit.

Answer 53

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

Answer 54

np