Question

A polynomial function h(x) with integer coefficients has a leading coefficient of -2 and a constant term of -1. According to the rational root theorem, which of the following are possible roots of h(x)?

a) 1
b) -1
c) -1/2
d) 1/2

Answer 1

According to the rational root theorem, the possible roots of a polynomial function h(x) with a leading coefficient of -2 and a constant term of -1 are ±1 or ±1/2. Thus, from the choices given, 1 and -1/2 are possible roots, while -1 and 1/2 are not.

Step-by-step explanation:

The rational root theorem states that if a polynomial function with integer coefficients has a rational number r/s as a root (where r and s are integers and have no common factors other than 1), then r must be a factor of the constant term and s must be a factor of the leading coefficient. In the case of the polynomial function h(x) with a leading coefficient of -2 and a constant term of -1, any rational root r/s must satisfy that r is a factor of -1, and s is a factor of -2.

The factors of -1 are ±1, and the factors of -2 are ±2, ±1. Therefore, the possible rational roots of the polynomial function h(x) could be ±1 or ±1/2. From the given choices: a) 1 and c) -1/2 are possible roots according to the rational root theorem, while b) -1 and d) 1/2 are not, because -1 is not a factor of the constant term, and 2 is not a factor of the leading coefficient.