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]↺↻]