Algebraic Expressions and Identities Notes
This chapter covers terms, coefficients, like and unlike terms, adding and subtracting expressions, multiplying monomials, binomials and polynomials, and the standard algebraic identities and how to use them.
- 3 min read
- 10 practice questions
- Aligned to CBSE 2025–26 syllabus
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- Updated Sept 2026
Use this map for a first read, then return to the checks and formulas during revision.
- 4 key formulas
- 1 concept checks
Terms, Coefficients and Types of Expressions
An algebraic expression is built from variables and constants using operations. Parts joined by + or − signs are terms; the number multiplying the variable part of a term is its coefficient.
Expressions are named by the number of terms: a monomial has one term (5x²), a binomial has two (x + 3), a trinomial has three (x² + 2x + 1); a polynomial is an expression with one or more terms and whole-number powers of the variables.
Like terms have exactly the same variable part (3xy and −7xy). Only like terms can be added or subtracted, by combining their coefficients.
Multiplying Expressions
Monomial × monomial: multiply the coefficients and multiply the variable parts using the exponent rule aᵐ × aⁿ = aᵐ⁺ⁿ. For example (3x²)(−4x³) = −12x⁵.
Monomial × polynomial: use the distributive law, multiplying the monomial into every term. For example 2x(x² − 3x + 5) = 2x³ − 6x² + 10x.
Binomial × binomial: multiply each term of the first by each term of the second and combine like terms. (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10.
Standard Identities
An identity is an equality true for every value of the variables (unlike an equation, which is true only for particular values).
The three basic identities are (a + b)², (a − b)², and (a + b)(a − b). A fourth, (x + a)(x + b) = x² + (a + b)x + ab, is very handy for products of binomials with a common variable.
Identities speed up expansion and factorisation, and help with mental arithmetic — for example 98 × 102 = (100 − 2)(100 + 2) = 100² − 2² = 9996.
Algebraic Expressions and Identities: revision sheet
Close the notes and explain each idea in your own words. Then check the formulas and attempt the examples before opening their solutions.
Recall checklist
- Terms, Coefficients and Types of Expressions
- Multiplying Expressions
- Standard Identities
For each topic: state the rule, explain when it applies, give an example and identify one mistake to avoid. Return to any topic you cannot explain without looking.
Formula recap (4)
Standard Identities
Standard Identities
Standard Identities
Standard Identities
Explain your method
Write the given information, choose a method, show the steps, then check units and whether the answer makes sense.
Check worked solution 1
Check worked solution 2
Turn mistakes into revision
Record the question, the step that went wrong and the corrected reason. Try it again tomorrow without the solution, then revisit it after a week. A correct guess still needs an explanation.
Practice and Revision
Test your understanding with quick chapter-level practice.
Chapter Q&A
What is the difference between an identity and an equation?
An equation is true only for particular value(s) of the variable (e.g. x + 2 = 5 only when x = 3). An identity is true for every value of the variable (e.g. (a + b)² = a² + 2ab + b²).
Can we add 3x and 3x²?
No. They are unlike terms because the powers of x are different. Only like terms (same variables with the same powers) can be combined.
How do identities help in mental calculation?
Numbers can be written as sums or differences of round numbers, then an identity turns a hard multiplication into easy squares or a difference of squares — e.g. 46 × 54 = (50 − 4)(50 + 4) = 2500 − 16 = 2484.