Simplifying Expressions

After expanding brackets, expressions often lie spread out everywhere: $x^2 + 3x + 2x + 6$ . Nothing wrong with it, just messy. Simplifying gathers the pieces so the expression wears its shortest outfit — and the gathering has one iron law in four words: only like merges with like.

Terms and factors: know your units

The chunks separated by plus and minus signs are called terms: $2x + 3 + 4x - 1$ has 4 terms — $2x$ , $3$ , $4x$ , $-1$ .

The pieces multiplied together inside a term are its factors: in the term $2x$ , the 2 and the $x$ are two factors. When merging, plus signs govern terms while times signs govern factors — two sets of rules, never to be mixed.

$3x + 2x = 5x$ : three $x$ s plus two $x$ s is five $x$ s — like terms, merge them;
$3x + 2y$ : $x$ and $y$ are different kinds, and 3 apples plus 2 bananas is still 3 apples and 2 bananas — unlike terms, stop right there.

Collecting like terms, three steps

Walk through $2x + 3 + 4x - 1$ :

Spot the like terms: $2x$ with $4x$ form one group, $3$ with $-1$ another;
Each group lines up: $2x + 4x = 6x$ , and $3 - 1 = 2$ ;
Write the simplest form: $2x + 3 + 4x - 1 = 6x + 2$ .

When terms swap positions, their signs travel with them: the minus on $-1$ belongs to it, and moving house without it is not allowed.

Multiplication simplifies too

Meeting a times sign, the rules change: factors multiply, and whatever can be computed gets computed.

$2 \times 3x = 6x$ — 2 and 3 are both numbers, multiply them first;
$x \times x = x^2$ — same base multiplied, exponents add;
$2x \times 3x = 6x^2$ — numbers times numbers, letters times letters, done in one move.

x + x + x is 3x, not x³

Addition counts copies; only multiplication raises powers. Three $x$ s added give $3x$ ; three $x$ s multiplied give $x^3$ . Writing $x + x + x$ as $x^3$ hands addition's job to multiplication.

Simplify first, then substitute

What is all this tidying for? Saving effort. When substituting, a simplified expression computes faster and steadier. Find the value of $2x + 3 + 4x - 1$ at $x = 5$ — both roads work:

Substitute directly: $2 \times 5 + 3 + 4 \times 5 - 1 = 10 + 3 + 20 - 1 = 32$ ;
Simplify first: $6x + 2 = 6 \times 5 + 2 = 32$ .

Same destination — simplifying changes the look, never the machine itself.

InteractiveSimplify Bench

A secret rule hides inside the machine. Feed it numbers and guess the rule!

? ? ?

Output

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Feed the machine the same number: $2x + 3 + 4x - 1$ and $6x + 2$ always spit out the same result. Simplifying is just giving the same machine a shorter name.

Check yourself

Quick quiz

1. Simplify 3x + 2y + 5x.
2. Simplify 2x + 3 + 4x − 1.
3. At x = 5, what is the value of 6x + 2?