m, n are zeroes of 2−5+. Find the value of a & c if their sum and product of zeroes is twice the negative coefficient of . PLEASE ANSWER PROPERLY WITH STEPS!! PLEASE HELP ME OUT!!
Answers
Answer:
In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies, in particular, that if the number of sign changes is zero or one, then there are exactly zero or one positive roots, respectively.
By a homographic transformation of the variable, one may use Descartes' rule of signs for getting a similar information on the number of roots in any interval. This is the basic idea of Budan's theorem and Budan–Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a list of disjoints intervals containing together all real roots of the polynomial, and containing each exactly one real root. Descartes rule of signs and homographic transformations of the variable are, nowadays, the basis of the fastest algorithms for computer computation of real roots of polynomials (see Real-root isolation).
Descartes himself used the transformation x → –x for using his rule for getting information of the number of negative roots.
Answer:
Descartes' Rule of Signs is a useful help for finding the zeroes of a polynomial, assuming that you don't have the graph to look at. This topic isn't so useful if you have access to a graphing calculator because, rather than having to do guess-n-check to find the zeroes (using the Rational Roots Test, Descartes' Rule of Signs, synthetic division, and other tools), you can just look at the picture on the screen. But if you need to use it, the Rule is actually quite simple.
Use Descartes' Rule of Signs to determine the number of real zeroes of:
f (x) = x5 – x4 + 3x3 + 9x2 – x + 5
Descartes' Rule of Signs will not tell me where the polynomial's zeroes are (I'll need to use the Rational Roots Test and synthetic division, or draw a graph, to actually find the roots), but the Rule will tell me how many roots I can expect, and of which type.