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?

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Jamari

Mathematics

7 January, 23:41

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?

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Darion Ingram · 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.