Terms, Expressions, and Equations

These three words come up constantly, and students mix them up all the time. So let's be really clear about what each one means:

  • A term is one chunk. It could be a number, a variable, or a number and a variable stuck together. Examples: 5, x, −3y², 4ab.
  • An expression is one or more terms added or subtracted together. No equals sign. Examples: 2x + 7, a² − 4b + 1.
  • An equation is when you set two expressions equal to each other. It always has an equals sign. Examples: 2x + 7 = 15, y = 3x − 2.

If a student says "I solved the expression," that's a red flag. There's nothing to solve without an equals sign. You simplify an expression. You solve an equation.

Term: −4xy
Expression: −4xy + 3x − 1
Equation: −4xy + 3x − 1 = 0

Identifying Coefficients

The coefficient is just the number sitting in front of the variable. It tells you how many of that variable-thing you've got. Getting this right matters, because you're about to start adding and subtracting them.

Term: 8m³ → coefficient is 8
Term: −y → coefficient is −1 (the 1 is invisible but present)
Term: x/3 → can be written ⅓x → coefficient is ⅓
Term: 6 → a constant term; coefficient discussion does not apply

When you multiply two terms, you multiply the coefficients and add the exponents: 3x² × 4x = 12x³. Multiply 3 × 4 to get 12, add the exponents 2 + 1 to get 3. That's the whole move.

Collecting Like Terms

Like terms have exactly the same variable part. Same letters, same exponents. If they match, you can add or subtract them. If they don't match, they stay separate as different terms in the expression. That's the whole rule, honestly.

Simplify: 9a − 3b + 2a + 7b − 5

Group by variable:
a terms: 9a + 2a = 11a
b terms: −3b + 7b = 4b
Constants: −5

Result: 11a + 4b − 5
Simplify: 4x² + 2x − x² + 3 − 5x

x² terms: 4x² − x² = 3x²
x terms: 2x − 5x = −3x
Constants: 3

Result: 3x² − 3x + 3

And don't try to combine x² and x. Different exponents means different terms. They can't be merged, even though they both have x in them.

Expanding Brackets

Expanding brackets just means getting rid of them. You take whatever's outside and multiply it by each thing inside, one at a time. Don't skip any terms.

Single bracket: 3(2x − 5)
= 3 × 2x + 3 × (−5)
= 6x − 15

Double bracket (FOIL): (x + 4)(x − 3)
First: x × x = x²
Outer: x × (−3) = −3x
Inner: 4 × x = 4x
Last: 4 × (−3) = −12
= x² − 3x + 4x − 12
= x² + x − 12

Here's a classic mistake: students expand (a − b)² and write a² + b². That's wrong. You have to actually multiply it out: (a − b)(a − b) = a² − 2ab + b². That middle term is what people miss. Use FOIL, don't just square each piece separately.

Factoring Out Common Terms

Factoring is just expanding in reverse. Instead of multiplying outward, you're pulling a common factor out front. Find the biggest thing that goes into every term, stick it outside the bracket, and divide each term by it. That factor is called the highest common factor (HCF).

Factor: 12x² + 8x
HCF of 12 and 8 = 4; HCF of x² and x = x
Common factor = 4x
12x² ÷ 4x = 3x; 8x ÷ 4x = 2
Result: 4x(3x + 2)

Check by expanding: 4x × 3x + 4x × 2 = 12x² + 8x ✓
Factor: 6a²b − 9ab²
HCF = 3ab
6a²b ÷ 3ab = 2a; 9ab² ÷ 3ab = 3b
Result: 3ab(2a − 3b)

Always check by expanding your answer back out. If you get what you started with, you got it right. If not, something went wrong with the HCF.