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