Re: Roots of polynomial equations
I like Serena said:
A third order polynomial has at least one real root, since it tends to different infinities in plus and minus directions.
If it has 2 real roots, one of those roots must be a duplicate root, meaning there are really 3 real roots.
This is not true.
For example, consider:
[math]p(x) = x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)[/math]
What IS true, is if a real cubic polynomial has 2 real roots, the third root must be "self-conjugate", and thus also real.
As MarkFL's post(s) indicate the sum of the squares of each root is a symmetric polynomial in each root, and so must be expressible in terms of elementary symmetric polynomials of the three roots. This symmetric polynomial is of degree 2, and so is expressible in terms of elementary polynomials of degree 2 or less...and we KNOW what these evaluate to...the coefficients of our original polynomial!
That is, if:
[math]x^3 + x + 12 = (x - r_1)(x - r_2)(x - r_3)[/math]
[math]= x^3 - (r_1+r_2+r_3)x^2 + (r_1r_2+r_2r_3+r_1r_3)x - r_1r_2r_3[/math]
Then:
[math]r_1+r_2+r_3 = 0[/math]
[math]r_1r_2+r_2r_3+r_1r_3 = 1[/math]
[math]r_1r_2r_3 = -12[/math].
So since [math]r_1^2 + r_2^2 + r_3^2 = (r_1+r_2+r_3)^2 - 2(r_1r_2+r_2r_3+r_1r_3)[/math],
[math]r_1^2 + r_2^2 + r_3^2 = 0^2 - 2(1) = -2[/math].
Since real squares are always non-negative, at least one (and thus at least two) of our roots must be non-real.
Then it *is* true that since a real cubic (by the opening remarks of ILikeSerena's post) always has at least one root, we know we have exactly one real root, and it is easy to show (see above) that this root lies between -3 and -2.
Taking [math]r_1 = \alpha[/math], and from the fact that [math]r_3 = \overline{r_2}[/math], we see from:
[math]r_1r_2r_3 = -12 = \alpha|r_2|^2 = \alpha|r_3|^2[/math]
that [math]-12 = \alpha|r_2|^2 < -2|r_2|^2[/math], so that:
[math]|r_2|^2 < \frac{-12}{-2} = 6 \implies |r_2| < \sqrt{6}[/math].
Similarly, from [math]-3|r_2|^2 < \alpha|r_2|^2 = -12[/math], we obtain:
[math]|r_2|^2 > \frac{-12}{-3} = 4 \implies |r_2| > 2[/math], or:
[math]2 < |r_2| = |r_3| < \sqrt{6}[/math].