The Rational Root Theorem (RRT) is a helpful tool in algebra that helps us find the roots of polynomial functions.

Let’s look at how it works!

Understanding the Rational Root Theorem

The RRT tells us that if we have a polynomial function like this:

$$f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$$

In this formula, $a_n$ is the first number (the leading coefficient), and $a_0$ is the last number (the constant). If this polynomial has a rational root (which means a fraction like $\frac{p}{q}$), then:

• $p$ (the top part) is a factor of the last number $a_0$.
• $q$ (the bottom part) is a factor of the first number $a_n$.

Steps to Find Rational Roots

1. Identify Coefficients: Let’s say we look at the polynomial $f(x) = 2x^3 - 3x^2 - 5x + 6$. Here, $a_n = 2$ and $a_0 = 6$.

2. Find Factors:

   • The factors of $a_0 = 6$ are $\pm 1, \pm 2, \pm 3, \pm 6$.
   • The factors of $a_n = 2$ are $\pm 1, \pm 2$.

3. List Possible Rational Roots: Using the factors of $6$ and $2$, we can combine them to see possible roots:

   • This gives us $\pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2}$.

Testing the Candidates

Instead of trying every possible number, we only need to test these likely candidates. For example, if we plug in $x = 1$ into our polynomial $f(x)$, we can calculate:

$$f(1) = 2(1)^3 - 3(1)^2 - 5(1) + 6 = 0.$$

Since $f(1) = 0$, this means $x = 1$ is a root!

Conclusion

The Rational Root Theorem makes finding rational roots much easier by letting us concentrate on a smaller list of possible values. This approach helps us work with polynomial functions in a more organized and quick way, allowing us to identify roots and continue our analysis smoothly.