Algebraic Expressions and Identities in Mathematics

Updated on November 1, 2025 | By Learnzy Academy

Algebraic Expressions
An algebraic expression is a combination of constants, variables, and mathematical operations like addition, subtraction, multiplication, and division.

Examples:
5x + 3
7a² - 4a + 6
2xy + 3y - 4

Terms, Coefficients, and Factors

  1. Term: Each part separated by + or – sign. Example: In 5x²y, 5x²y is one term.
  2. Coefficient: The numerical part of the term. Example: In 5x²y, coefficient is 5.
  3. Variables: The letters used in an expression. Example: x and y.
  4. Factors: The parts multiplied together. Example: 5, x², and y are factors of 5x²y.

Types of Algebraic Expressions

  1. Monomial: Expression with one term. Example: 7x, 3a²b
  2. Binomial: Expression with two terms. Example: a + b, 2x - 5y
  3. Trinomial: Expression with three terms. Example: x² + 2x + 1
  4. Polynomial: Expression with one or more terms.

Common Algebraic Identities:

  1. (a + b)² = a² + 2ab + b²
  2. (a - b)² = a² - 2ab + b²
  3. (a + b)(a - b) = a² - b²
  4. (a + b)³ = a³ + 3a²b + 3ab² + b³
  5. (a - b)³ = a³ - 3a²b + 3ab² - b³
  6. a³ + b³ = (a + b)(a² - ab + b²)
  7. a³ - b³ = (a - b)(a² + ab + b²)

Example 1: Expand (x + 3)²
(x + 3)² = x² + 2×3×x + 3² = x² + 6x + 9

Example 2: Simplify (a + b)(a - b)
(a + b)(a - b) = a² - b²

Applications

  1. Simplifying algebraic expressions
  2. Solving equations
  3. Factorizing polynomials
  4. Finding area and perimeter using formulas
Click here to download practice questions on Algebraic Expressions and Identities

List of question on "Algebraic Expressions and Identities"

  1. Factorize completely: a^4 – 4a²b² + 4b^4
  2. Calculate the volume of a cuboidal box whose dimensions are 5x, 3x², and 7x^4.
  3. Shiv works in a mall and gets paid Rs 50 per hour. Last week he worked for 7 hours and this week he will work for x hours. Write an algebraic expression for the money paid to him for both the weeks.
  4. Rohan's mother gave him Rs 3xy² and his father gave him Rs 5(xy² + 2). Out of this total money, he spent Rs (10 – 3xy²) on his birthday party. How much money is left with him?
  5. What should be subtracted from 2x³ – 3x²y + 2xy² + 3y³ to get x³ – 2x²y + 3xy² + 4y³?
  6. How much is 21a³ – 17a² less than 89a³ – 64a² + 6a + 16?
  7. Arjun bought a rectangular plot with length x and breadth y and then sold a triangular part of it whose base is y and height is z. Find the area of the remaining part of the plot.
  8. Sonu and Raj have to collect different kinds of leaves for a science project. They go to a park where Sonu collects 12 leaves and Raj collects x leaves. After some time, Sonu loses 3 leaves and Raj collects 2x more leaves. Write an algebraic expression to find the total number of leaves collected by both of them.
  9. Amisha has a square plot of side m and another triangular plot with base and height each equal to m. What is the total area of both plots?
  10. Using identities, evaluate 8.56 × 11.60
  11. The area of a rectangle is uv, where u is the length and v is the breadth. If the length is increased by 5 units and the breadth is decreased by 3 units, find the new area of the rectangle.
  12. The cost of one pen is (x + 3) rupees and the cost of one pencil is (x – 2) rupees. Find the total cost of 4 pens and 5 pencils.
  13. Simplify: [(a + b)² – (a – b)²] ÷ (2ab)
  14. Simplify: (x + y)² + (y + z)² + (z + x)² – (x² + y² + z²)
  15. Factorize: (x² + 2x + 1) – (y² + 2y + 1)
  16. If the sum of two numbers is (a + b) and their difference is (a – b), find the product of the numbers.
  17. The side of a square is (x + 3). Find the expression for its area.
  18. Find the value of (3x + 4)² when x = 2.
  19. Simplify: (p + q)² – (p – q)²
  20. Add: (3x² + 5x + 7) and (4x² – 2x + 3)
  21. Find the value of 2x + 3y when x = 4 and y = 5.
  22. Identify the coefficient of x² in the expression 5x² + 3x + 7
  23. Subtract 5xy(x + y − 5) from x(6x² − 7y + 5) + xy(x + y)
  24. What must be added to the sum of (x² − 4x + 7) and (2x² + 5x − 9) to get 0?
  25. Rohan purchased a rectangular plot whose two adjacent sides are (y – 6x + 32 + 8) and (x – 2y – 5z – 8). He wants to put a wire fence twice around it. Find the total length of wire needed.
  26. What will be the product if we multiply double of (x − 2/x) by triple of (x + 2/x)?
  27. What should be subtracted from 3a + 7b – 10 to get –2a – 7b + 9?
  28. Determine the product of (3a + 2b) and (9a² – 6ab + 4b²).
  29. What is the sum of ab, a + b, and b + ab?
  30. Simplify 7x²(3x – 9) + 3 and find its value for x = 4 and x = 6.
  31. Find the value of x, if 10000x = (9982)² – (18)²
  32. Find the value of (38² − 22²) / 16 using a suitable identity.
  33. Verify that (11pq + 4q)² – (11pq – 4q)² = 176pq²
  34. Multiply (x² + 2y) by (x³ – 2xy + y³) and find the value of the product for x = 1 and y = –1.
  35. Multiply (6x² – 5x + 3) by (3x² + 7x – 3)
  36. Multiply (3x² + 5y²) by (5x² – 3y²)
  37. Subtract: (4x + 5) from (−3x + 7)
  38. Add: 8x² + 7xy – 6y², 4x² – 3xy + 2y² and -4x² + xy – y²