Big picture
An algebraic expression uses numbers, variables and operations to describe a quantity. An identity is an equality that remains true for every permitted value of its variables. These tools let you organise patterns, calculate efficiently and represent practical situations without knowing every number in advance.
If one notebook costs ₹x, the cost of five notebooks and a ₹12 pen is 5x + 12 rupees. Multiplying or simplifying this expression changes its form while keeping its value. This chapter explains why the rules work, not merely which symbols to memorise.
Learning goals
Identify terms, factors, coefficients, constants and types of expressions.
Distinguish like terms, unlike terms, expressions, equations and identities.
Add, subtract and multiply expressions using the distributive law.
Derive and apply the standard square and product identities.
Evaluate expressions and use identities for numerical calculation.
Model area and cost problems while preserving units and restrictions.
Core ideas
1. Variables, constants, terms and coefficients
A variable represents a number that may vary; a constant has a fixed value. Terms are the parts joined by addition or subtraction at the outermost level. In 6x² − 4xy + 9, the terms are 6x², −4xy and 9. Retain the sign when naming a term.
A factor is multiplied to form a term. The factors of −4xy include −4, x and y. The numerical coefficient of xy is −4. In −p, the coefficient of p is −1, not zero. Coefficients can be understood relative to a chosen factor: in 7ab, the coefficient of a is 7b.
Worked example. For 5m²n − 8n + 11, list the terms as 5m²n, −8n, 11. The numerical coefficients of the variable terms are 5 and −8, and the constant term is 11. The exponent belongs to its variable: m²n is not mn².
2. Like terms and classifications
Like terms have identical variable factors with identical exponents. Their numerical coefficients may differ. Thus 3x²y and −8x²y are like, whereas 3xy² is unlike either. The order of multiplication does not matter: 4ab and 7ba are like terms.
Expression type | Number of nonzero unlike terms after simplification | Example |
|---|---|---|
Monomial | One | −6p²q |
Binomial | Two | x + 5 |
Trinomial | Three | a² + 2ab + b² |
Polynomial | A finite sum of permitted monomial terms | 2x³ − x + 4 |
In the polynomial setting, variable exponents are nonnegative integers; 1/x and √x are not polynomial terms in x. Classify after combining terms: x + 2x = 3x, a monomial. A monomial is also a polynomial; these categories are not mutually exclusive.
3. Expressions, equations and identities
An expression has no equality to solve: 3x + 2. An equation asserts equality and may hold only for particular values: 3x + 2 = 14 gives x = 4. An identity holds for every allowed value: 3(x + 2) = 3x + 6.
Worked check. At x = 4, 3x + 2 = 3 × 4 + 2 = 14. At x = 1, it equals 5, so the equation 3x + 2 = 14 is not an identity. Checking a few values can disprove a claimed identity, but cannot prove that it holds for all values; algebraic derivation supplies the proof.
4. Addition and subtraction
Combine only like terms by adding their coefficients. Unlike terms remain separate. For (4a + 3b − 6) + (2a − 5b + 9), collect a terms, b terms and constants: (4 + 2)a + (3 − 5)b + (−6 + 9) = 6a − 2b + 3.
Subtraction applies to every term in the subtracted expression. Worked example: subtract 3p − 2q + 4 from 8p + q − 7. Write (8p + q − 7) − (3p − 2q + 4). Remove brackets: 8p + q − 7 − 3p + 2q − 4. Combine: 5p + 3q − 11. The word “from” determines the order.
5. Multiply monomials
Multiply numerical coefficients using sign rules. For powers of the same base, add exponents: xᵐ × xⁿ = xᵐ⁺ⁿ. Keep different variables separate. A negative times a negative is positive.
Worked example: (−3a²b)(4ab³) = (−3 × 4)(a² × a)(b × b³) = −12a³b⁴. Do not multiply exponents to obtain a²b³ or combine a and b into one variable.
For an area example, a rectangle with length 7xy cm and breadth 3x cm has area (7xy)(3x) = 21x²y cm², assuming positive lengths. Algebraic factors carry the same unit logic as ordinary measurements.
6. Multiply a monomial and a polynomial
The distributive law states a(b + c) = ab + ac. Every term inside the bracket must be multiplied. It also handles subtraction: a(b − c) = ab − ac.
Worked example: −2t(3t² − 4t + 5) = (−2t)(3t²) + (−2t)(−4t) + (−2t)(5) = −6t³ + 8t² − 10t. The middle term is positive because both factors are negative.
For two polynomials, multiply every term in one by every term in the other, then combine like terms. (x + 4)(x − 7) = x² − 7x + 4x − 28 = x² − 3x − 28. With a trinomial, keep the same method: (y + 2)(y² − y + 3) = y³ − y² + 3y + 2y² − 2y + 6 = y³ + y² + y + 6.
7. Square of a sum
Expand (a + b)² as (a + b)(a + b): a² + ab + ab + b². Therefore (a + b)² = a² + 2ab + b². The middle term appears twice, so simply adding the separate squares is incorrect.
A diagram can show a square of side a + b divided into one a² square, two ab rectangles and one b² square. Their areas add to the total. For a and b positive, this gives a geometric explanation of the identity.
Worked example: (2x + 5)² = (2x)² + 2(2x)(5) + 5² = 4x² + 20x + 25. Treat 2x as the whole first quantity, so its square is 4x².
8. Square of a difference
Expand (a − b)(a − b) to obtain (a − b)² = a² − 2ab + b². The final term remains positive because (−b)(−b) = b².
Worked example: (3p − 4q)² = (3p)² − 2(3p)(4q) + (4q)² = 9p² − 24pq + 16q². A useful check is p = q = 1: the original gives (3 − 4)² = 1, and the expansion gives 9 − 24 + 16 = 1.
9. Product of a sum and difference
Multiplication gives (a + b)(a − b) = a² − ab + ab − b². The middle terms cancel, yielding (a + b)(a − b) = a² − b².
Worked example: (5m + 2n)(5m − 2n) = (5m)² − (2n)² = 25m² − 4n². This is different from (5m − 2n)², which would have a middle term. Recognise the pair of opposite signs before selecting the identity.
10. Products with a common first term
The identity (x + a)(x + b) = x² + (a + b)x + ab follows from multiplying x² + bx + ax + ab. It is useful when two binomials share x but have different constants.
Worked example: (z − 3)(z + 8) = z² + (−3 + 8)z + (−3)(8) = z² + 5z − 24. The sign of the product is negative because the constants have opposite signs. Substitute z = 2 to check: (−1)(10) = −10 and 4 + 10 − 24 = −10.
11. Numerical evaluation and applications
Choose an identity matching a nearby convenient number. Worked example: 103² = (100 + 3)² = 10000 + 600 + 9 = 10609. Similarly, 98² = (100 − 2)² = 10000 − 400 + 4 = 9604. For 204 × 196, use (200 + 4)(200 − 4) = 40000 − 16 = 39984.
Substitution requires brackets around negative values. At x = −2 and y = 3, 2x² − 3xy = 2(−2)² − 3(−2)(3) = 8 + 18 = 26. Squaring −2 gives 4; writing −2² without clear grouping can cause a sign error.
A square garden of side x metres is extended by 3 metres along both dimensions. Added area = (x + 3)² − x² = x² + 6x + 9 − x² = 6x + 9 m². At x = 7, added area = 42 + 9 = 51 m²; directly, 10² − 7² = 100 − 49 = 51. The calculation assumes x is positive because it represents a length.
A reliable approach to questions
Identify terms, signs, variables and the required operation.
For addition or subtraction, remove brackets correctly and collect like terms.
For multiplication, distribute every term and apply sign and exponent rules.
Select an identity only when its full pattern matches.
Simplify, then check using a convenient substitution or a second calculation.
In word problems, state the quantity, restrictions and final unit.
Connections that matter
Factorisation reverses these expansions and identities.
Mensuration uses algebraic expressions for lengths, perimeters and areas.
Linear equations use simplified expressions on either side of an equality.
Common mistakes
Adding unlike terms: 3x + 2y cannot become 5xy.
Forgetting subtraction signs: change every sign in the subtracted bracket.
Omitting 2ab in a square: expand two brackets to recover both middle products.
Using a² − b² for a difference squared: opposite-sign factors and identical factors are different patterns.
Revision checklist
I can identify terms, factors and coefficients.
I can combine like terms and subtract expressions correctly.
I can multiply monomials and polynomials.
I can derive and apply the standard identities.
I can evaluate expressions with negative substitutions.
I can solve area and numerical problems with units and checks.