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?

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.