Find the upper and lower bounds..

Question

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation.

 

*I need answer in order to check with the answer i got, thank you!

 

x⁴ − x³ − 2x² + 3x + 5 = 0

 

upper bound is ____

lower bound is ____

Answer 1

there are 4 potential real solutions, given a 4th degree polynomial

using Descartes Rule of Signs there is a maximum 2 positive real solutions (the number of sign changes in the coefficients of the terms, including the constant term)

and maximum 2 negative real solutions (count the sign changes in the coefficients after changing x to -x)

imaginary solutions come in conjugate pairs

there are either 2 real + solutions and 2 - real solutions

or

2 real + and 2 imaginary

or

2 real - and 2 imaginary

or

4 imaginary

upper bound is 4 for real solutions

lower bound is 0 for real solutions