How can I use the Rational Root Theorem to find all possible rational roots for the equation 3x³ + 9x + 6 = 0?

The Rational Root Theorem provides a way to identify all potential rational roots of a polynomial equation. To apply this theorem to the polynomial equation 3x³ + 9x + 6 = 0, follow these steps:

1. Identify the coefficients:

   The polynomial can be expressed as:

   • a₃ = 3 (coefficient of x³)
   • a₂ = 0 (coefficient of x², there is no x² term)
   • a₁ = 9 (coefficient of x)
   • a₀ = 6 (constant term)
2. Find the factors of the constant term:

   The constant term is 6. The factors of 6 are:

   • ±1
   • ±2
   • ±3
   • ±6
3. Find the factors of the leading coefficient:

   The leading coefficient is 3. The factors of 3 are:

   • ±1
   • ±3
4. Determine all possible rational roots:

   Using the Rational Root Theorem, the possible rational roots are all the combinations of factors of the constant term over the factors of the leading coefficient.
   Thus, the possible rational roots can be calculated as:

   • ±1/1 = ±1
   • ±2/1 = ±2
   • ±3/1 = ±3
   • ±6/1 = ±6
   • ±1/3 = ±1/3
   • ±2/3 = ±2/3
   • ±3/3 = ±1
   • ±6/3 = ±2

Thus, the complete list of possible rational roots is:

• ±1
• ±2
• ±3
• ±6
• ±1/3
• ±2/3

Please note that while these are the potential rational roots, some of them may not be actual roots of the equation. You will need to test each one by substituting them back into the original equation or using synthetic division to verify.