Question

According to the rational root theorem, which of the following are possible roots of the polynomial function below? f(x) = 6x^3 - 7x^2 + 8

A) 4
B) 3
C) 2/3
D) -8

Answer 1

Utilizing the Rational Root Theorem, the possible rational roots of the polynomial f(x) = 6x^3 - 7x^2 + 8 include 4 and 2/3 from the given options.

Step-by-step explanation:

The Rational Root Theorem helps us determine the possible rational roots of a polynomial equation. For the polynomial f(x) = 6x^3 - 7x^2 + 8, the possible rational roots are the factors of the constant term (8) divided by the factors of the leading coefficient (6). Factoring these numbers, we get the set of potential roots that includes ± 1, ± 2, ± 4, ± 8, and ± 1/6, ± 1/3, ± 2/3, ± 1/2.

Looking at the given options:

• 4 (A) is a potential root because 4 is a factor of 8.
• 3 (B) is not a potential root because it is not a factor of the constant term 8 divided by the leading coefficient 6.
• ±2/3 (C) is a potential root because 2/3 is one of the possible factors when dividing the factors of 8 by those of 6.
• ±-8 (D) is a potential root because -8 is a factor of 8.

From the given options, 4 and 2/3 are possible rational roots according to the Rational Root Theorem.