Remember rational numbers? They're just numbers that can be written as a fraction, like $1/2$ or $-7/3$. The word "rational" comes from "ratio." It's one integer divided by another.

Now, let's bring algebra into the mix. Instead of just integers, what if we used polynomials?

When you create a fraction with a polynomial in the numerator and another in the denominator, you get a rational expression.

Rational expressions are all around in algebra. They can look simple or complex, but they all share the same structure: a polynomial divided by another polynomial.

The last two examples are not rational expressions because their numerators are not polynomials. Polynomials can't have variables inside square roots or have negative exponents on their variables.

There's one crucial rule for all fractions, and it applies to rational expressions, too: the denominator can never, ever be zero. Division by zero is undefined in mathematics.

For a simple fraction like $7/x$, the rule is straightforward: $x$ cannot be 0. But for a more complex expression, we need to be more careful. Consider this expression:

Here, the value of the numerator doesn't matter for this rule. We only care about the denominator, $x-4$. We must ensure it isn't zero.

This value, $x=4$, is called an excluded value. It's a value that the variable is not allowed to be. For any other value of $x$, the expression is perfectly valid.

Identifying these excluded values is the first step before performing any other operations on rational expressions, like adding, subtracting, or simplifying them.

Let's test your understanding of these new concepts.

Understanding what makes an expression rational and how to find its excluded values is the foundation for everything that comes next.