Ken Ward's Mathematics Pages Descartes Rule of Signs

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Descartes' Rule of Signs

Descartes Rule of Signs can be used to determine the number of positive and negative real roots of a polynomial.
I proceed by examples.

Example 1

Suppose we have an equation:
f(x)= +x^{4}-3x^{3}-15x^{2}+19x+30 [1]
I have put a + sign before the highest power above for clarification.
This equation has the signs: + - - ++
It has 2 sign changes, so the equation has 2 or (multiples of 2 less)
positive real roots. That is, it has 2 or 0 positive real roots.
We can check the number of negative real roots by noting:
f(-x)=+x^{4} -3x^{3}-15x^{2}-19x+30
The signs are:
+ - - - +
That is, there are 2 sign changes, so the equation has 2 (or multiples
of 2 less) negative real roots. Or, 2 or 0 negative real roots.
Looking at the original equation [1], we can note the signs for the
negative case by reversing the sign of all the odd powers to get the
same answer.

Example 2

Consider:
x^{3}-2x^{2}-13x-10
The signs are:
+ - - - (the first plus comes from the
x^{3}term)
There is one sign change, so there is one real positive root. When the
indicated number of roots is an odd number, we can be sure that there
is one real root of that kind, in this case, a positive reeal root.
For the negative roots, either reversing the signs of the odd powers, or computing f(-x):
- - + -
There are 2 sign changes, so the equation has 2 or 0 negative roots.
The roots are -1, -2, and 5

Example 3

x^{4}-37x^{2}+24x+180
The signs are:
+ - + +
So there 2 or 0 positive real roots
For the negative roots, the signs are:
+ - - +
There are 2 sign changes, so there are 2 or 0 negative real roots
The roots are: -2, -6, 3, 5

Example 4

x^{5}-4x^{4}-3x^{3}+28x^{2}+60x+15
The signs for the positive root are:
+ - - + + +
There are 2 sign changes, so the equation has 2 or 0 real positive roots.

For the negative roots:
- - + + - +
There are also 3 sign changes and therefore 3 or 1 real negative roots.
The roots of this equation are:
Root0= 3.63998984 – 1.92654117i
Root1= 3.63998984 +1.92654117i
Root2= – 1.49501719 +0.90281543i
Root3= – 0.28994531
Root4= – 1.49501719 – 0.90281543i
There is one negative real root.

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