EDB — 1D7

[1D7]Let \(p(x) = a_ n x^ n + a_{n-1} x^{n-1} + . . . + a_ 0\) a polynomial with all real roots and coefficients all non-zero. Show that the number of positive roots (counted with multiplicity) is equal to the number of sign changes in the sequence of coefficients of \(p\). (Hint. Use induction on \(n\), using the fact that between two consecutive roots of \(p\) there exists a root of \(p'\).) This result is known as Descartes’ rule of signs. [ [1D8]↺↻]