Question

According to the rational root theorem which of the following are possible routes of the polymerase function below ?​

f(x)=6x³-7x²+2x+8
a.2/3
b.-8
c.4
d.3
e.-1/6
f.3/4

Answer 1

According to the Rational Root Theorem, the possible rational roots of f(x) = 6x³ - 7x² + 2x + 8 are ratios of the factors of the constant term and the leading coefficient. Therefore, the possible roots from the given options are 2/3, -8, 3, and -1/6. Options 4 and 3/4 are not valid according to the theorem.

Step-by-step explanation:

The Rational Root Theorem provides a way to list all possible rational roots of a polynomial equation. According to this theorem, the possible rational roots of the polynomial f(x) = 6x³ - 7x² + 2x + 8 are the ratios of the factors of the constant term (in this case 8) to the factors of the leading coefficient (in this case 6).

The factors of 8 are ±1, ±2, ±4, and ±8. The factors of 6 are ±1, ±2, ±3, and ±6.

Therefore, all possible rational roots are ±1, ±2, ±3, ±4, ±6, ±8, ±1/2, ±3/2, ±1/3, ±2/3, ±1/6, and ±4/6 (which simplifies to ±2/3).

Reviewing the options given, we see that:

• 2/3 is a possible root because both 2 and 3 are factors of the constant term and leading coefficient, respectively.
• -8 is a possible root because it is a factor of the constant term.
• 3 is a possible root because it is a factor of the constant term.
• -1/6 is a possible root because 1 and 6 are factors of the constant term and leading coefficient, respectively.

The options 4 and 3/4 are not possible roots of the function because 4 is not a factor of 6 and 3/4 does not meet the requirement of the Rational Root Theorem.