+124 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 theorem 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 the 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?
Hi, thank you for reading this :)). Could you please help me
+121 Given the information provided, Manu can narrow down the list of possible rational roots using the Rational Root Theorem and the given bounds for the polynomial. The Rational Root Theorem states that if a rational number \(p/q\) is a root of a polynomial with integer coefficients, then \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient.
From the information given, we know that there are 3 roots to the equation. This means that the polynomial is a cubic polynomial.
Descartes' Rule of Signs indicates that there are no sign changes in \(P(x)\) and 3 sign changes in \(P(-x)\). This implies that there are 3 positive roots and no negative roots. This aligns with the fact that the lower bound of the polynomial is -6 and the upper bound is 1.
Considering all these facts, we can deduce that the roots of the polynomial are likely to be rational numbers from the given list that are positive. So, Manu should try the following values from the list of possible rational roots in Synthetic Division:
- \(1/4\)
- \(1/2\)
- \(1\)
These values align with the information obtained from the Fundamental Theorem of Algebra, Descartes' Rule of Signs, and the given bounds of the polynomial.
