To find the possible rational roots of the polynomial equation x³ + x² + x + 3 = 0, we can apply the Rational Root Theorem. This theorem states that any potential rational root, in the form of a fraction p/q, consists of factors of the constant term divided by factors of the leading coefficient.
In this equation:
- The constant term (the term without x) is 3.
- The leading coefficient (the coefficient of the highest degree term, x³) is 1.
Now, let’s find the factors:
- Factors of 3: ±1, ±3
- Factors of 1: ±1
Using these factors in the Rational Root Theorem, the possible rational roots are:
- p/q = ±1/1 yields: ±1
- p/q = ±3/1 yields: ±3
So, the complete list of possible rational roots for the polynomial equation x³ + x² + x + 3 = 0 is:
- 1
- -1
- 3
- -3
This means the potential rational roots that you can test for the equation are 1, -1, 3, and -3.