To find the possible rational roots of the polynomial equation 3x³ + 9x + 6 = 0, we can utilize the Rational Root Theorem. This theorem states that any rational solution, in the form of a fraction p/q, must have the numerator p as a factor of the constant term (the term without x) and the denominator q as a factor of the leading coefficient (the coefficient of the highest degree term).
For our equation:
- Constant term = 6
- Leading coefficient = 3
First, we identify the factors of the constant term, 6:
- Factors of 6: ±1, ±2, ±3, ±6
Next, we identify the factors of the leading coefficient, 3:
- Factors of 3: ±1, ±3
Now, using the Rational Root Theorem, the possible rational roots are determined by taking the factors of the constant term divided by the factors of the leading coefficient:
- Possible rational roots = ±1/1, ±2/1, ±3/1, ±6/1, ±1/3, ±2/3, ±3/3, ±6/3
This simplifies down to:
- ±1, ±2, ±3, ±6, ±1/3, ±2/3
So, the complete list of possible rational roots for the equation 3x³ + 9x + 6 = 0 is:
- 1, -1, 2, -2, 3, -3, 6, -6, 1/3, -1/3, 2/3, -2/3