To find the possible rational roots of the polynomial equation x³ + x² + x + 3 = 0, we can utilize the Rational Root Theorem. This theorem states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient.
In our equation, the constant term is 3 and the leading coefficient (the coefficient of x³) is 1.
Step 1: Identify the factors.
- Factors of the constant term (3): ±1, ±3
- Factors of the leading coefficient (1): ±1
Step 2: List the possible rational roots.
According to the Rational Root Theorem, the possible rational roots (p/q) can be formed by taking each factor of the constant term and dividing them by each factor of the leading coefficient:
- Possible roots: ±1, ±3 (because ±1 is the only factor of the leading coefficient)
Thus, the possible rational roots of the polynomial equation x³ + x² + x + 3 = 0 are: 1, -1, 3, -3.