Algebraic Expressions
1. Basic Terminology
| Term | Definition | Example |
|---|---|---|
| Variable | A symbol (letter) that can take DIFFERENT values | x, y, z |
| Constant | A quantity with a FIXED value | 5, -3, 2/7 |
| Term | A constant, a variable, or their PRODUCT | 3x, -5y², 7 |
| Coefficient | The NUMERICAL factor of a term | In 3xy, coefficient = 3 |
| Algebraic Expression | A combination of terms connected by + or — | 3x² — 5x + 7 |
'In the term —4x²y, the coefficient is —4. The literal factors are x, x, and y.'
2. Types of Expressions
| Type | Definition | Example |
|---|---|---|
| Monomial | ONE term | 5x²y, -3ab |
| Binomial | TWO terms | 2x + 3y, a² — b² |
| Trinomial | THREE terms | x² + 2x + 1 |
| Polynomial | MANY terms (general) | x³ + 2x² — 3x + 4 |
Degree of a polynomial: The HIGHEST power of the variable.
Degree of 4x³ — 2x² + x — 7 is 3. Degree of a²b + b²c (sum of exponents: 2+1 = 3) is 3.
3. Like and Unlike Terms
Like terms: Have the SAME literal factors (same variables with same exponents). Unlike terms: Have DIFFERENT literal factors.
Like terms: 3x²y, —5x²y, ½x²y (all have x²y) Unlike terms: 3x²y, 3xy² (different exponents)
'Only LIKE terms can be added or subtracted. Unlike terms CANNOT be combined.'
4. Addition and Subtraction of Algebraic Expressions
Method: Combine ONLY like terms by adding or subtracting their coefficients.
Worked Example: Add 3x² + 2xy — 5 and x² — 4xy + 3.
(3x² + 2xy — 5) + (x² — 4xy + 3) = 3x² + x² + 2xy — 4xy — 5 + 3 = 4x² — 2xy — 2
Worked Example: Subtract (2x² — 5x + 3) from (5x² — 2x + 7).
(5x² — 2x + 7) — (2x² — 5x + 3) = 5x² — 2x + 7 — 2x² + 5x — 3 = 3x² + 3x + 4
5. Multiplication of Algebraic Expressions
Monomial × Monomial
Multiply coefficients, then multiply variables.
(—3x²y) × (4xy³) = (—3 × 4) × (x² × x) × (y × y³) = —12x³y⁴
Monomial × Polynomial
Use DISTRIBUTIVE law: Multiply the monomial with EACH term.
2x(3x² — 4x + 5) = 6x³ — 8x² + 10x
Binomial × Binomial
Multiply each term of one with each term of the other.
(x + 2)(x + 5) = x(x + 5) + 2(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10
Worked Example: Multiply (3x — 2y)(2x + 3y)
= 3x(2x + 3y) — 2y(2x + 3y) = 6x² + 9xy — 4xy — 6y² = 6x² + 5xy — 6y²
Binomial × Trinomial
Each term of the binomial multiplies each term of the trinomial.
Worked Example: (x + 3)(x² — 2x + 5)
= x(x² — 2x + 5) + 3(x² — 2x + 5) = x³ — 2x² + 5x + 3x² — 6x + 15 = x³ + x² — x + 15
6. Division of Algebraic Expressions
Monomial ÷ Monomial
Divide coefficients and subtract exponents of like variables.
12x⁵y³ ÷ 3x²y = (12/3) × x⁵⁻² × y³⁻¹ = 4x³y²
Polynomial ÷ Monomial
Divide each term of the polynomial by the monomial.
(6x³ — 9x² + 12x) ÷ 3x = 2x² — 3x + 4
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Adding coefficients of unlike terms' | Only COMBINE like terms. 3x + 2y CANNOT be combined |
| '—(x — y) = —x — y' | —(x — y) = —x + y. The sign of EACH term inside changes |
| 'x² × x³ = x⁶' | x² × x³ = x⁵ (ADD exponents, not multiply) |
| 'Forgetting the sign when subtracting' | When subtracting, change ALL signs of the subtracted expression |
ICSE Exam Focus (5–7 marks)
- 2-mark questions: Identify type of expression, degree, coefficient
- 3-mark questions: Add or subtract algebraic expressions
- 4-mark questions: Multiply binomials and trinomials
- 6-mark questions: Combined operations — simplify complex expressions
Self-Test
Q1. Classify as monomial, binomial, or trinomial: 3x²y, x + y, a² + b² + c². A1. 3x²y → monomial. x + y → binomial. a² + b² + c² → trinomial.
Q2. Add: (2a² — 3ab + b²) + (a² + 2ab — 3b²) + (—a² + ab + b²) A2. a² terms: 2a² + a² — a² = 2a². ab terms: —3ab + 2ab + ab = 0. b² terms: b² — 3b² + b² = —b². Answer: 2a² — b².
Q3. Multiply: (2x — 3y)(3x + 2y) A3. = 2x(3x+2y) — 3y(3x+2y) = 6x² + 4xy — 9xy — 6y² = 6x² — 5xy — 6y².
Q4. Simplify: (x + 2)(x² — 3x + 1) A4. = x(x²—3x+1) + 2(x²—3x+1) = x³ — 3x² + x + 2x² — 6x + 2 = x³ — x² — 5x + 2.
Q5. Divide: 15x⁴y³z² ÷ (—5x²yz) A5. = (15/—5) × x⁴⁻² × y³⁻¹ × z²⁻¹ = —3x²y²z.
Q6. Subtract (3x² — 4x + 7) from (5x² — 2x — 3). A6. (5x² — 2x — 3) — (3x² — 4x + 7) = 5x² — 2x — 3 — 3x² + 4x — 7 = 2x² + 2x — 10.
