The Rational Root Theorem provides a way to identify possible rational roots for a polynomial equation. According to this theorem, if a polynomial $f(x)$ has integer coefficients, then any rational root $\frac{p}{q}$ of the polynomial must satisfy:

• $p$ is a divisor of the constant term.
• $q$ is a divisor of the leading coefficient.

In this case, the polynomial $f(x)$ has:

• Leading coefficient = $-12$
• Constant term = 1

Step 1: Divisors of the constant term (1)

The divisors of 1 are $\pm 1$.

Step 2: Divisors of the leading coefficient (-12)

The divisors of $-12$ are: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$

Step 3: Possible rational roots

The possible rational roots are all fractions of the form $\frac{p}{q}$, where $p$ is a divisor of 1 and $q$ is a divisor of $-12$. Therefore, the possible rational roots are:

$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}$

Step 4: Checking the given values

The given possible roots are: $-3, -1, 12, 6$.

Let's check each one:

• $-3$: Not a possible root because it is not of the form $\pm \frac{1}{q}$.
• $-1$: Possible root because $-1$ is $p = \pm 1$.
• $12$: Not a possible root because it is not of the form $\pm \frac{1}{q}$.
• $6$: Not a possible root because it is not of the form $\pm \frac{1}{q}$.

Conclusion

The only possible root from the list provided is $-1$.

Would you like more details on how the Rational Root Theorem works, or do you have other questions?

Here are 5 related questions:

1. How do you apply the Rational Root Theorem to a polynomial with different degrees?
2. Can the Rational Root Theorem guarantee that all rational roots of a polynomial will be found?
3. How do you determine if a possible rational root is actually a root of the polynomial?
4. What is the process for factoring a polynomial using the Rational Root Theorem?
5. How can you find the irrational roots of a polynomial?

Tip: If you're ever unsure about a possible root, test it by directly substituting it into the polynomial to see if it results in zero!