7. Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

A rational expression is any polynomial divided by any polynomial except for zero. If your students don't already know, dividing by zero is catastrophic. In fact, many esteemed mathematicians believe the apocalypse will result from dividing by zero.

Students should understand that rational expressions are closed under addition, subtraction, multiplication, and division, meaning that:

• A rational expression plus a rational expression is a rational expression
• A rational expression minus a rational expression is a rational expression
• A rational expression times a rational expression is a rational expression
• A rational expression divided by a rational expression is a rational expression

It's only rational.

Students should also know how to add, subtract, multiply, and divide rational expressions. They're really not that different from adding, subtracting, and multiplying polynomials. We're just throwing division in there, too.

Let's denote a rational expression as 

We could worry about remainders here, but keeping rational expressions as fractions will help us out in the long run. In fact, rational expressions and fractions are practically identical when it comes to adding, subtracting, multiplying, and dividing.

If we want to add or subtract ¹⁄₄ and ²⁄₅, we first have to find a common denominator. Only after they both have a denominator of 5 × 4 = 20 can we add and subtract them. We don't need common denominators to multiply ¹⁄₄ and ²⁄₅, though. Just multiply the numerators together and multiply the denominators together. When division is involved, it's easier to just flip the second fraction and then multiply, so ¹⁄₄ ÷ ²⁄₅ becomes ¹⁄₄ × ⁵⁄₂.

Well guess what. Rational expressions are exactly the same, only the integers are replaced with polynomials. Basically, the rules simplify to these:

In each case, we end up with a polynomial divided by a polynomial, which is a rational expression. Since polynomials can be defined anywhere, we can't have any x under the radical. As long as students know how to multiply and factor polynomials, they should be fine with rational expressions.