How many negative roots has the equation
$\displaystyle x^6 - bx^5 - 2ax^3 - cx + a^2=0, b\geq0, c\geq0?$
Keep Smiling
Malay
That would be true, Cap'n. I forgot about that. In Descartes rule, it is assumed that terms with 0 coefficients are deleted and the constant term is not 0. Also, descartes rule says the number of negative roots is equal to the change in signs or less than that by an even integer.
Let's say b and c are 0, then we'd have $\displaystyle (-x)^{6}+2a(-x)^{3}+a^{2}=x^{6}-2ax^{3}+a^{2}$
2 change of signs. Has 2 negative roots or 0 negative roots.
If b and c were not 0 , then from before. No change of signs. No negative zeros.
BTW, Quick, $\displaystyle (-x)^{6}=x^{6}$. May I ask where you got the negative from?. Maybe I am missing something.
First I believe Descartes rule does not work for these polynomials. To work IOriginally Posted by malaygoel
believe that it requires all the powers less than the maximum appear with non-
zero coefficients. So it applies to:
$\displaystyle x^7+x^6+2x^5-x^4-x^3-x^2+x-1$,
where the signs are +,+,+,-,-,-,+,-, which change three times, and so this
polynomial has at most three positive roots, and so has either 1 or 3 positive
roots.
Now for the negative roots we switch the signs of the odd powers to give:
$\displaystyle -x^7+x^6-2x^5-x^4+x^3-x^2-x-1$,
now the signs are -,+,-,-,+,-,-,-, which change sign 4 times, so there
are at most four negative roots to the original polynomial. So the original
polynomial has 4, 2 or 0 negative roots.
RonL
You will note the what we call weasel words in my earlier post aboutOriginally Posted by Malay
the applicability of the rule of signs to polynomials with missing powers.
(weasel words - wording which alows the author to subsequently disavow
what they wrote).
It appears that the rule of signs is applicable . I have looked at the
problem that I thought might exist with such polynomials again and
it turns out they are not real so we can proceed:
$\displaystyle
x^7+x^6-x^4-x^3-x^2+x-1
$
Has signature +,+,-,-,+,-. The signs chenge from + to - or - to + three
times in this signature so there are at most three positive roots.
Now to investigate the negative roots we change the signs of all the
odd power terms in the polynomial and then proceed as before:
$\displaystyle
-x^7+x^6-x^4+x^3-x^2-x-1
$
which has signature -,+-,+,-,-. The signs change 4 times in this signature,
so there are at most four negative roots.
RonL