The Rational Root Theorem is a powerful tool used in algebra to identify possible rational roots of polynomial equations. To apply this theorem to the polynomial equation x3 + x2 + x + 3 = 0, we begin by identifying its coefficients and looking at the structure of the polynomial.
In our case, the polynomial can be expressed as:
x3 + 1x2 + 1x + 3
The Rational Root Theorem states that any potential rational root, in the form of a fraction p/q, where:
- p is a factor of the constant term (in this case, 3), and
- q is a factor of the leading coefficient (in this case, the coefficient of x3, which is 1).
Now, let’s identify the factors:
- Factors of the constant term (3): ±1, ±3
- Factors of the leading coefficient (1): ±1
Next, we can find all the possible rational roots by forming all combinations of these factors:
- Possible rational roots (p/q):
- ±1 (from p = ±1 and q = ±1)
- ±3 (from p = ±3 and q = ±1)
Thus, the complete list of possible rational roots for the polynomial equation x3 + x2 + x + 3 = 0 is:
- x = 1
- x = -1
- x = 3
- x = -3
In summary, according to the Rational Root Theorem, the potential rational roots for the equation are ±1 and ±3.