The French mathematician René Descartes (1596–1650) recognized a connection between the roots of a polynomial equation and the
Let P(x) be a polynomial with real coefficients written in standard form.
In both cases, count multiple roots according to their multiplicity.
What does Descartes’ Rule of Signs tell you about the real roots of
There are two sign changes,
Therefore, there are either 0 or 2 positive real roots.
Why can't there be zero negative real roots?
The number of negative roots is equal to 1 or is less than 1 by an even number. Zero is less than 1 by an odd number.
Recall that graphs of cubic functions have zero or two turning points. Because the graph already shows two turning points, it will not change direction again. So there are no positive real roots.
Use the Rational Root Theorem to list all possible rational roots for each equation.
Write a polynomial function with rational coefficients so that
Reasoning In the statements below, r and s represent integers. Is each statement always, sometimes, or never true? Explain.