Question

Manu determines the roots of a polynomial equation p(x)=0 by applying the theorems he knows. He organizes the results of these theorems. From the fundamental theorems of algebra, Manu knows there are 3 roots to the equation. From Descartes' rule of sign, Manu finds no sign changes in p(x) and 3 sign changes in p(-x). The rational root theorem yields +1/4,+1/2,+1,+5/4,+2,+5/2,+4,+5,+10,+20 as a list of possible rational roots. The lower bound of the polynomial is -6. The upper bound of the polynomial is 1. What values in Manu's list of rational roots should he try in synthetic division in light of these findings?

Answer 1

Manu should considered only the possible rational zeros between -6 and 1, because those are the lower and upper bonds. A lower bond means that possible rational zero is not less than -6, and the higher number possible is not more than 1. This way, Manu has a interval to try. This rational method with bonds is aimed to narrow all possible solutions.

So, if Manu find out through the synthetic division that possible roots are +1/4,+1/2,+1,+5/4,+2,+5/2,+4,+5,+10,+20, then he only should considered those inside the intervals marked by the lower and upper bonds, which are +1/4,+1/2,+1, because the rest is higher than 1.

Therefore, Manu should try first +1/4,+1/2 and +1.