Rational Expressions — Core Skills

Rational Expressions: The Basics

Part 1 of 2 — One Skill, One Idea

A rational expression is a fraction with letters in it, like $\frac{3x + 12}{x + 4}$. The word "rational" here means ratio, which means fraction. Nothing more.

Two rules run this whole topic.

Rule 1: the bottom can never be zero

Dividing by zero is not allowed in math. So any value of $x$ that makes the bottom equal $0$ is thrown out. That value is called undefined, or excluded from the domain.

To find it, set the bottom equal to $0$ and solve. For $\frac{x + 1}{x - 8}$, set $x - 8 = 0$ and add to both sides: . So is excluded.

Rule 2: you may cancel factors, not terms

A factor is something being multiplied. A term is something being added or subtracted. You may only cancel a whole factor that appears on the top and on the bottom.

Worked example

Simplify $\frac{3x + 12}{x + 4}$.

Step 1 — Look at the top. Both $3x$ and $12$ are divisible by $3$, so factor it out: $3x + 12 = 3(x + 4)$.

Step 2 — Rewrite the fraction: $\frac{3(x + 4)}{x + 4}$.

Step 3 — Now $(x + 4)$ is a factor on the top and a factor on the bottom, so they cancel. Anything divided by itself is $1$.

Step 4 — What is left is $3$.

So $\frac{3x + 12}{x + 4} = 3$.

The mistake to avoid

In $\frac{x + 4}{x + 6}$ you cannot cross out the two $x$ letters. They are terms being added, not factors being multiplied. That fraction is already as simple as it gets.

Rational Expressions: Practice

Part 2 of 2 — Run the Steps

To simplify a rational expression:

1. Factor the top. Pull out a common number, or use the two-number trick, or spot a difference of squares.
2. Factor the bottom the same way.
3. Cancel any factor that appears on both. A factor is something multiplied, never something added.
4. Write what is left.

To find where an expression is undefined:

1. Set the bottom equal to $0$.
2. If the bottom is already factored, set each parenthesis equal to $0$ separately.
3. Solve. Every answer you get is a value $x$ cannot be.

Two patterns that show up a lot

Difference of squares on top. $$ factors into , because . If the bottom is , that factor cancels and is left.