Rene Descartes

Rene Descartes (1596-1960) is a famous French mathematician and philosopher. He also is a physician, mechanic, physiologist, the founder of modern analytic geometry and algebraic symbolic, the author of the method of radical doubt in philosophy, the author of mechanistic in physics, the creator of forerunner in reflexology. He formulated the following statement in 1637 year – the theory (or the rule) of Descartes.

Image Source

Rene Descartes born in 1956 year in the town Lae. His mother died when he was just one year old. The father of Rene was a judge so he wasn’t in Lae too often. The grandmother of Rene was growing him up. Rene was very unhealthy and curious.

The theorem (the rule) of Descartes

Image Source

The number of positive roots of polynomial doesn’t exceed the number of signs’ changes in the sequence of its coefficients and differs from it by an even number. That’s time to explain the proof of this theorem (rule).

The proof

Image Source

Let’s assume that without loss of generality. Let’s hold an induction on the number of sighs’ changes. If in the sequence A there’re not signs’ changes, all non-zero numbers of this sequence are positive, so for any and the polynomial f(x) doesn’t have the positive roots. It’s the base of induction.
Let’s suppose now that the theorem is already proved for all polynomials that have a less number of signs’ changes in the sequence of coefficients than q. Let’s suppose that is a polynomial in which sequence coefficients are equal q changes of signs and c is a positive roof of this polynomial.
Then:

where’s

i.e.

This equation implies that:

Let’s prove that there’re less signs’ changes in the sequence B than in the sequence A. Let’s suppose that p – is a number of signs’ changes in sequence B. Let’s notice now that
If l is the first place of the signs’ change in sequence B , the numbers b_0 and b_1 have different signs. If that’s so, what means that the signs of numbers are different. So that, the sequence contents at least one change of sign.
Let’s suppose that l and k are two neighboring places of signs’ change in the sequence B. If it’s like that, numbers have different signs too. Consequently, there’s a change of sign too among the numbers