Question

the number 3 is an upper bound for the set of roots of this polynomial function. f(x)=3x^4-5x^3-5x^2+5x+2 (true or false?)

Answer 1

Use the rational root theorem.

Possible real values of x are defined as +/- P/Q. P is a set containing all factors of the CONSTANT of the polynomial whereas Q is a set containing all factors of the LEADING COEFFICIENT of the polynomial.

Answer 2

2 is your constant, 3 is your leading coefficient. Both are prime, so P = {1, 2} and Q = {1, 3}.

You can have four possible combinations of these: {1/1, 1/3, 2/1, 2/3} simplifying to {1/3, 2/3, 1, 2}. This makes for eight total possible rational roots including negatives.

The number 3 is not contained in the set P/Q, so 3 cannot be an upper bound of the set.

Answer 3

thanks I understand it now