What is Descarte rule of sins?

Question

Please show an example.

Answer 1

Descartes' rule of signs (note spellings) is a way of determining, with minimal calculation, the maximum number of positive real roots and negative real roots that a given polynomial has. See sources for a complete explanation and examples of use.

Answer 2

Descartes' rule of signs, first described by René Descartes in his work La Geometrie, is a technique for determining the number of positive or negative roots of a polynomial.

The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2. Multiple roots of the same value are counted separately.

As a corollary of the rule, the number of negative roots is the number of sign changes after negating the coefficients of odd-power terms (otherwise seen as substituting the negation of the variable for the variable itself), or less than it by a multiple of 2.

For example, the polynomial
x³ +x² -x -1
has one sign change between the second and third terms. Therefore it has exactly 1 positive root.

Negating the odd-power terms gives
-x³ +x² +x -1

This polynomial has two sign changes, so the original polynomial has 2 or 0 negative roots.

The polynomial factors easily as

(x + 1)² (x-1)

so the roots are -1 (twice) and 1.

Answer 3

You mean of Descartes' rule of signs.

"Given a polynomial such as:

x^4 + 7x^3 - 4x^2 - x - 7

it is possible to say anything about how many positive real roots it has, just by looking at it?
Here's a striking theorem due to Descartes in 1637, often known as "Descartes' rule of signs": The number of positive real roots of a polynomial is bounded by the number of changes of sign in its coefficients. Gauss later showed that the number of positive real roots, counted with multiplicity, is of the same parity as the number of changes of sign" (1)
and check the reference (2)

Answer 4

Descarte's rule of signs says that the number of positive real roots of a polynomial is bounded by the number of changes of sign in its coefficients.