Algebraic Expressions
1. What Is an Algebraic Expression?
An ALGEBRAIC EXPRESSION is a combination of CONSTANTS and VARIABLES connected by +, -, ×, and ÷.
Examples: 3x + 2y, 5a² - 3ab + 7, 2x + 3y - 4z.
Components
- Variable: A letter whose value can change (x, y, a, b, ...).
- Constant: A number with a fixed value (5, -3, 1/2, ...).
- Term: A part of the expression separated by + or -.
Types of Expressions
| Type | Number of Terms | Example |
|---|---|---|
| Monomial | 1 term | 5x, -3ab, 7 |
| Binomial | 2 terms | 3x + 2y, a² - b² |
| Trinomial | 3 terms | 2x² - 3x + 5 |
| Polynomial | 1 or more terms | All of the above |
2. Terms, Coefficients, and Like/Unlike Terms
Terms and Coefficients
In 4x²y - 3xy + 7y - 5:
| Term | Variable Part | Numerical Coefficient |
|---|---|---|
| 4x²y | x²y | 4 |
| -3xy | xy | -3 |
| 7y | y | 7 |
| -5 | none (constant) | -5 |
Like and Unlike Terms
- Like terms: SAME variable(s) raised to the SAME power(s).
- 3x² and -7x² are LIKE terms.
- 4xy and 5yx are LIKE terms (xy = yx).
- Unlike terms: Different variables or different powers.
- 3x and 3x² are UNLIKE terms.
- 2xy and 3xz are UNLIKE terms.
Important
Only LIKE terms can be added or subtracted.
3. Addition and Subtraction of Algebraic Expressions
Method 1: Horizontal Addition
(3x² - 2x + 5) + (5x² + 3x - 7) = 3x² + 5x² - 2x + 3x + 5 - 7 = 8x² + x - 2.
Method 2: Column Addition
Write like terms in the SAME column.
3x² - 2x + 5
+ 5x² + 3x - 7
------------
8x² + 1x - 2
Subtraction
'Add the NEGATIVE of the second expression.' (5x² - 3x + 2) - (3x² + x - 4) = 5x² - 3x + 2 - 3x² - x + 4 = 2x² - 4x + 6.
Worked Example (ICSE 2024, 2 marks)
Subtract: (2x² - 3xy + y²) from (5x² + 2xy - 3y²).
Solution: = (5x² + 2xy - 3y²) - (2x² - 3xy + y²) = 5x² + 2xy - 3y² - 2x² + 3xy - y² = 3x² + 5xy - 4y².
4. Multiplication of Algebraic Expressions
Monomial × Monomial
Multiply coefficients. Add exponents of same bases. Example: (3x²y)(-2xy³) = 3 × (-2) × x²⁺¹ × y¹⁺³ = -6x³y⁴.
Monomial × Binomial (Distributive Law)
a(b + c) = ab + ac. Example: 2x(3x + 4y) = 6x² + 8xy.
Monomial × Trinomial
a(b + c + d) = ab + ac + ad. Example: -3a(2a² - 4a + 1) = -6a³ + 12a² - 3a.
Binomial × Binomial
Use FOIL method or DISTRIBUTIVE law. (a + b)(c + d) = ac + ad + bc + bd.
Example (ICSE 2023, 3 marks): Simplify: (2x + 3)(3x - 2). = 2x(3x - 2) + 3(3x - 2) = 6x² - 4x + 9x - 6 = 6x² + 5x - 6.
Special Product (ICSE Focus)
(x + a)(x + b) = x² + (a + b)x + ab. Example: (x + 5)(x - 3) = x² + (5 - 3)x + 5(-3) = x² + 2x - 15.
5. Finding the Value of an Expression
Substitute the given value of the variable(s) and simplify.
Example (ICSE 2024, 2 marks)
'Find the value of 3x² - 2xy + y² when x = 2, y = -1.' = 3(2)² - 2(2)(-1) + (-1)² = 3(4) + 4 + 1 = 12 + 4 + 1 = 17.
6. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Identifying terms, coefficients | 1-2 marks | Low |
| Addition and subtraction | 2-3 marks | Very High |
| Multiplication (all types) | 3-4 marks | Very High |
| Finding value of expression | 2 marks | High |
| Word problems to form expressions | 2-3 marks | Medium |
Common Mistakes
- Adding/subtracting UNLIKE terms (e.g., 3x + 2y = 5xy — WRONG).
- Sign errors: subtracting a negative term.
- Forgetting to multiply the coefficient (e.g., 2x × 3x = 5x — WRONG, it is 6x²).
- Wrong exponent addition: x² × x³ = x⁶ (WRONG — it is x⁵).
Self-Test (5 Questions)
Q1. Identify the number of terms in 5x²y - 3xy + 2y - 7. (1 mark)
Q2. Add: (3a² - 5ab + 2b²) + (-2a² + 7ab - b²). (2 marks)
Q3. Multiply: (-2xy²)(3x²y³). (2 marks)
- A) -6x³y⁵
- B) 6x³y⁵
- C) -6x²y⁶
- D) 6x²y⁶
Q4. Simplify: (3x - 2)(2x + 5). (3 marks)
Q5. Find the value of 4a² - 3ab + 2b² when a = 2, b = -3. (2 marks)
Answers
A1. 4 terms. A2. a² + 2ab + b². A3. A) -6x³y⁵. ((-2)(3)x¹⁺²y²⁺³ = -6x³y⁵.) A4. 6x² + 11x - 10. (3x(2x+5) - 2(2x+5) = 6x² + 15x - 4x - 10 = 6x² + 11x - 10.) A5. 16 + 18 + 18 = 52. (4(4) - 3(2)(-3) + 2(9) = 16 + 18 + 18 = 52.)
