Important Formulas Algebraic Expressions - (Maths) Class 7 (Old NCERT)

Algebraic expressions use letters and numbers together to represent quantities and relationships. The letters (called literals or variables) stand for numbers, while numerical symbols that do not change are called constants. This section collects the essential definitions, rules and formulae that students must know when working with algebraic expressions.

Basic terms and their meanings

• Literal numbers / Literals are letters that stand for numbers.
• Variable is a symbol that can take on different numerical values.
• Constant is a symbol with a fixed numerical value.
• Base and index (exponent): In x⁹, the number 9 is the index or exponent and x is the base. In a⁵, 5 is the index and a is the base.
• Algebraic expression is formed by combining constants and variables using addition, subtraction, multiplication and division.
• Term is each addend of an expression. For example, the terms 4xy and 7 make the expression 4xy + 7.
• Factor is a quantity that multiplies with other quantities to give a term. In the term 4xy, the factors are 4, x and y. Factors that contain variables are called algebraic factors.
• Coefficient is the numerical part of a term. Any factor of a term may be treated as the coefficient of the remaining part.
• Polynomial is an expression with one or more terms. A one-term polynomial is a monomial, a two-term polynomial is a binomial, and a three-term polynomial is a trinomial.
• Like terms are terms that have the same algebraic (variable) part. For example, 4xy and -3xy are like terms. Unlike terms have different variable parts, for example 4xy and -3x.
• The degree of a term is the sum of the exponents of the variables in that term. The degree of a polynomial is the highest degree among its terms.
• Zero polynomial is the polynomial equal to 0; its terms are all zero and its degree is not usually defined.
• The sum or difference of like terms is another like term whose coefficient is the sum or difference of the coefficients.
• To subtract one expression from another, change the sign of each term in the expression to be subtracted and then add the two expressions.
• If a grouping symbol (bracket) is preceded by a negative sign and the bracket is removed, the sign of each term inside the bracket changes from '+' to '-' and '-' to '+'.

Rules and formulae for manipulating expressions

• Commutative laws:
  a + b = b + a
  a × b = b × a
• Associative laws:
  (a + b) + c = a + (b + c)
  (a × b) × c = a × (b × c)
• Distributive law:
  a(b + c) = ab + ac
• Multiplication of monomials:
  (ax^m)(bxⁿ) = ab x^m+n
• Power of a product:
  (ab)ⁿ = aⁿ bⁿ
• Power of a power:
  (a^m)ⁿ = a^mn
• Special products:
  (a + b)² = a² + 2ab + b²
  (a - b)² = a² - 2ab + b²
  a² - b² = (a - b)(a + b)
• Factorisation by common factor:
  If each term of an expression has a common factor g, then g may be taken outside:
  g(a + b + c) = ga + gb + gc
• Combining like terms:
  ax + bx = (a + b)x

Solved examples with stepwise solutions

Example 1. Combine like terms: 4xy + 7 + 3xy - 2

Write the original expression.

4xy + 7 + 3xy - 2

Group like terms together (pair terms having the same variable part).

(4xy + 3xy) + (7 - 2)

Add the coefficients of like terms and simplify the constants.

7xy + 5

Answer: 7xy + 5

Example 2. Subtract the expression (2x - 5y) from (4x + 3y).

Write the subtraction as an addition after changing the sign of each term in the subtracted expression.

(4x + 3y) - (2x - 5y) = (4x + 3y) + (-2x + 5y)

Combine like terms.

(4x - 2x) + (3y + 5y)

Simplify the coefficients.

2x + 8y

Answer: 2x + 8y

Example 3. Expand (x + 3)² using a special product formula.

State the special product formula.

(a + b)² = a² + 2ab + b²

Substitute a = x and b = 3.

x² + 2·x·3 + 3²

Simplify the multiplication and the square.

x² + 6x + 9

Answer: x² + 6x + 9

Example 4. Factorise 6x + 9 by taking out the greatest common factor.

Identify the greatest common factor (GCF) of 6x and 9.

GCF = 3

Take 3 outside the bracket.

3(2x + 3)

Answer: 3(2x + 3)

Practical tips and common errors

• Always identify like terms by comparing the variable part exactly (same variables raised to the same powers).
• When removing brackets preceded by a negative sign, change the sign of each term inside the bracket.
• Keep coefficients and variable parts clearly separate; do not combine unlike terms (e.g., x and x² are unlike).
• Use the distributive law to expand products and to factor expressions by taking common factors.
• When working with exponents, add exponents when multiplying like bases and multiply exponents when raising a power to a power.

These definitions, rules and worked examples form the core set of formulas and procedures used for algebraic expressions at the school level. Regular practice of combining like terms, expanding using distributive law and special products, and factoring by common factors will build fluency in algebra.