Roots of the equation x^3 - 3x - m(m^3) = 0

To find the roots of the equation x^3 - 3x - m(m^3) = 0, we need to solve for the values of x that satisfy the equation. The equation is a cubic equation, which means it can have up to three roots.

Using the Rational Root Theorem

The Rational Root Theorem states that if a rational number p/q is a root of a polynomial equation with integer coefficients, then p must be a factor of the constant term and q must be a factor of the leading coefficient.

In this equation, the constant term is -m(m^3) and the leading coefficient is 1. The possible rational roots are therefore the factors of -m(m^3) divided by the factors of 1.

Identifying the Possible Rational Roots

To find the factors of -m(m^3), we can break it down into its prime factors. The prime factorization of -m(m^3) is -1 * m * m * m * m.

The factors of -1 are 1 and -1.
The factors of m are 1 and m.
The factors of m^3 are 1, m, m^2, and m^3.

Therefore, the possible rational roots are:

1, -1, m, -m, m^2, -m^2, m^3, -m^3

Testing the Possible Rational Roots

Using these possible rational roots, we can substitute them into the equation x^3 - 3x - m(m^3) = 0 and check if they satisfy the equation. If a value satisfies the equation, then it is a root.

For example, if we substitute x = m into the equation, we get:

(m)^3 - 3(m) - m(m^3) = 0
m^3 - 3m - m(m^3) = 0
m^3 - 3m - m^4 = 0

Similarly, we can substitute the other possible rational roots and check if they satisfy the equation.

Finding the Actual Roots

By substituting the possible rational roots into the equation, we can determine which values satisfy the equation and are therefore the roots of the equation x^3 - 3x - m(m^3) = 0.

The actual roots of the equation can vary depending on the value of m. Therefore, the roots of the equation x^3 - 3x - m(m^3) = 0 are represented by the options:

A) m, m^3
B) m^3, -m
C) -m, -(m^3)

The specific values of m that correspond to these roots can be determined by substituting the values into the equation and solving for m.