The Rational Root Theorem is a useful technique for identifying potential rational roots of a polynomial equation.

According to the Rational Root Theorem, if a polynomial equation with integer coefficients has a rational root, that root can be expressed in the form $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient. This implies that to determine the possible rational roots of a polynomial, we should first list all the factors of the constant term and all the factors of the leading coefficient. We can then create all possible fractions by combining one factor from each list.

For instance, consider the polynomial equation

Here, the constant term is $-2$, which has factors of $\pm 1$ and $\pm 2$. The leading coefficient is $2$, which also has factors of $\pm 1$ and $\pm 2$. Consequently, the potential rational roots of this equation are formed by the combinations of these factors:

These fractions simplify to the possible roots: $\pm 1$, $\pm \frac{1}{2}$, $\pm 2$, and $\pm 1$.

To utilize the Rational Root Theorem for solving a polynomial equation, we can test each of the identified potential rational roots by substituting them into the equation to see if the result equals zero. If we discover a root that satisfies this condition, we can apply polynomial division to factor the equation and uncover the remaining roots. If none of the potential rational roots yield a solution, we may need to employ alternative methods, such as the quadratic formula or numerical techniques, to find the roots.