Algebraic Expressions
CBSE Class 8 – Mathematics
1. Simple Explanation
| Term | Meaning |
|---|---|
| Expression | A mathematical phrase that contains variables, numbers and operations ( +, – , ×, ÷, ^ ). |
| Variable | A letter that represents an unknown or changing value (e.g., x, y, z). |
| Constant | A fixed value (e.g., 5, –3). |
| Term | A single number or a product of a number and one or more variables (e.g., (3x^2), (–5yz)). |
| Coefficient | The number that multiplies the variable(s) in a term (e.g., 3 in (3x^2)). |
| Like terms | Two terms that have the exact same variables raised to the same powers (e.g., (4x^2) and (-2x^2)). |
| Unlike terms | Terms that are not like terms (different variables or different powers). |
Algebraic expressions are the building blocks of algebra; they let us symbolise quantities and relationships that change.
2. Key Points
| Concept | Key Facts |
|---|---|
| Writing an expression | Write terms in any order; usually arranged from highest to lowest degree of the variable(s). |
| Combining like terms | Add or subtract only like terms. Sum of coefficients becomes the new coefficient. |
| Simplifying | 1. Combine like terms. 2. Use exponent rules: (a^m \cdot a^n = a^{m+n}). 3. Distribute multiplication over addition. |
| Evaluating an expression | Substitute the given value of the variable(s) and compute. |
| Special products | 1. ((a + b)^2 = a^2 + 2ab + b^2) 2. ((a - b)^2 = a^2 - 2ab + b^2) 3. ((a + b)(a - b) = a^2 - b^2) |
| Powers of variables | (x^0 = 1), (x^1 = x). Exponents must be whole numbers (non‑negative, except when explicitly allowed). |
| Negative coefficients | Keep the sign with the coefficient (e.g., (-4x)). |
| Zero term | 0 has no variable part; it can be added or subtracted freely. |
| Distributive property | (a(b + c) = ab + ac). Use it to expand products of binomials or to factor. |
| Factoring a common factor | If a term appears in every part of an expression, factor it out. Example: (3x^2 + 6x = 3x(x + 2)). |
3. Worked Example
Problem
Simplify the expression
[ 3x^2y - 5xy^2 + 2x^2y + 4xy^2 - 7xy ]
and then evaluate it for (x = 2) and (y = -1).
Step 1: Identify like terms
| Term | Type | Coefficient |
|---|---|---|
| (3x^2y) | (x^2y) | 3 |
| (-5xy^2) | (xy^2) | –5 |
| (2x^2y) | (x^2y) | 2 |
| (4xy^2) | (xy^2) | 4 |
| (-7xy) | (xy) | –7 |
Step 2: Combine like terms
- (x^2y) terms: (3x^2y + 2x^2y = 5x^2y)
- (xy^2) terms: (-5xy^2 + 4xy^2 = -xy^2)
- (xy) term remains as (-7xy)
So the simplified expression is
[ \boxed{5x^2y - xy^2 - 7xy} ]
Step 3: Evaluate for (x = 2,; y = -1)
Substitute:
[ 5(2)^2(-1) - (2)(-1)^2 - 7(2)(-1) ]
Compute each part:
- (5(4)(-1) = 5 \times (-4) = -20)
- (- (2)(1) = -2)
- (- 7(2)(-1) = -7 \times (-2) = 14)
Add them:
[ -20 - 2 + 14 = -8 ]
[ \boxed{,\text{Result} = -8,} ]
4. Quick Revision Summary
| Topic | What to remember |
|---|---|
| Terms & Coefficients | A term = coefficient × variable(s). Coefficient is the number in front. |
| Like & Unlike Terms | Combine only like terms; unlike terms stay separate. |
| Simplification | 1. Combine like terms. 2. Use exponent rules. 3. Distribute multiplication. |
| Evaluation | Replace variables with given numbers, then compute. |
| Special Products | ((a±b)^2), ((a+b)(a-b)=a^2-b^2). |
| Factoring | Pull out common factors: (ax + ay = a(x + y)). |
| Distributive Property | (a(b+c) = ab + ac). |
| Zero Term | 0 can be added/sub |