Find upper and lower bounds

Question

find integers that are upper and lower bounds for real zero polynomials

P(x)=x^4-2x^3+x^2-9x+2

Answer 1

actual real zeros = x intercepts = about .2 and 2.7

-2 is lower bound

2 is the upper bound if there were rational zeros

take integer factors of the constant term divided by integer factors of the coefficient of the highest degree term

by Descartes rule of signs there are 0, 2 or 4 positive real zeros and 0 negative real zeros. 0 would be the lower bound as there are no negative zeros

x f(x) = x^4 -2x^3 +x^2 -9x +2

0 2

1 1-2+1--+2 -9 +2 = -5

2 16 -16 +4 -18+2 = -12

3 81 -54 +9 -27 +2 = 11

at least one zero is between x=2 and 3, since f(x) changes sign from - to +

3 is the upper bound for real zeros