List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). (Enter your answers as a comma-separated list.)
S(x) = 6x4 − x2 + 6x + 12
To find the possible rational zeros of a polynomial, we can use the Rational Zeros Theorem. According to the theorem, the possible rational zeros of a polynomial are all the possible combinations of the factors of the constant term (the term without any variables) over the factors of the leading coefficient (the coefficient of the highest power of x).
In the given polynomial S(x) = 6x^4 - x^2 + 6x + 12, the constant term is 12, and the leading coefficient is 6.
To find the factors of the constant term, we can list all the possible integers that can divide 12. These are: ±1, ±2, ±3, ±4, ±6, and ±12.
To find the factors of the leading coefficient, we can list all the possible integers that can divide 6. These are: ±1, ±2, ±3, and ±6.
Combining these factors, we get the possible rational zeros: ±1, ±2, ±3, ±4, ±6, and ±12.
So, the list of possible rational zeros, according to the Rational Zeros Theorem, is: -12, -6, -4, -3, -2, -1, 1, 2, 3, 4, 6, and 12.