Algebraic Expressions — Recall & Practice

🎯 The Golden Idea

RECALL TRICK: “Same Family → Combine. Different Family → Keep.”

Terms can be added/subtracted only when their variable part is exactly the same.

3x + 5x = 8x but 3x + 5y cannot be combined

Coefficient

Number multiplying the variable.

7x → 7

Variable

Letter representing a number.

7x → x

Term

Parts separated by + or −.

4x² − 3x + 7

Constant

A term without a variable.

1. Like and Unlike Terms

Like terms

Same variables with the same powers.

3x², −7x², ½x²

Unlike terms

Different variable and/or different power.

3x², 3x, 3y

Memory check: Ignore the coefficient. Compare the letter + power.

Terms	Like?	Reason
5x, −2x	✓	Both have x¹
4a², 9a	✗	Power is different
7xy, −3xy	✓	Both have xy
2ab, 2a	✗	Missing b

2. Addition & Subtraction

“Keep the variable part; operate on the coefficients.”

3x + 7x = (3 + 7)x = 10x

9a − 4a = (9 − 4)a = 5a

5x² + 2x − 3x² + 7x = 2x² + 9x

Do NOT: add exponents when adding terms.
x² + x² = 2x², not x⁴.

When a bracket is preceded by +

5x + (3x − 2) = 5x + 3x − 2 = 8x − 2

When a bracket is preceded by −

5x − (3x − 2) = 5x − 3x + 2 = 2x + 2

Memory trick: “+ bracket = signs stay; − bracket = signs flip.”

3. Opening Brackets

DIS +: “The sign outside belongs to EVERY term inside.”

Positive outside

+(a + b − c) = a + b − c

Signs stay the same.

Negative outside

−(a + b − c) = −a − b + c

Every sign flips.

Number multiplied by a bracket

3(2x − 5) = 6x − 15

Distribute the multiplier to every term.

Variable multiplied by a bracket

x(3x + 4) = 3x² + 4x

Common error: 3(x + 2) ≠ 3x + 2. The 3 must multiply both terms.

4. Multiplication of Algebraic Expressions

MEMORY TRICK: “Coefficient × coefficient; variable × variable.”

(3x)(4x) = 12x²

(−2a)(5b) = −10ab

2x(3x − 4) = 6x² − 8x

(x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15

FOIL for two binomials: First • Outside • Inside • Last.
(a+b)(c+d)=ac+ad+bc+bd

Useful identities

(a+b)² = a² + 2ab + b²

(a−b)² = a² − 2ab + b²

(a+b)(a−b) = a² − b²

5. Division of Algebraic Fractions

BIG TRICK: “KEEP → CHANGE → FLIP.”
For division by a fraction: keep the first fraction, change ÷ to ×, flip the second fraction.

3x4y ÷ 9x²8y³ = 3x4y × 8y³9x² = 2y²3x

Reliable cancellation rule

Cancel only factors, not terms separated by + or −.

Allowed:
6x²3x = 2x

Not allowed:
x+3x ≠ 4

Power law reminder

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

So x⁵ ÷ x² = x³.

Best workflow: Factor → Flip (if division) → Cancel → Multiply → Simplify.

6. “Spot the Mistake”

❌ 3(x + 4) = 3x + 4

Correct: 3x + 12. The 3 distributes to both terms.

❌ −(2x − 5) = −2x − 5

Correct: −2x + 5. Every sign inside flips.

❌ x² + x³ = x⁵

Wrong because these are unlike terms. They cannot be combined.

❌ (x + 2)/x = 2

Terms cannot be cancelled across addition. You may factor first only when a common factor exists.

❌ 2x × 3x = 6x

Correct: 6x². Exponents add when multiplying the same base.

7. Interactive Practice

Try the question before revealing the answer. The questions change when you press New Question.

8. Teacher's 30-Second Recall

Like terms?
Same variable + same power.

Add/Subtract?
Combine coefficients only.

− bracket?
Flip every sign.

Multiply bracket?
Distribute to every term.

Fraction ÷ fraction?
Keep → Change → Flip.

Cancel?
Factors only, never separate terms.