Vardaan
Class 7 Maths • Chapter 12

Algebraic Expressions

Vardaan Learning Institute • Detailed Notes with Practice Questions

🔤 1. Key Terms — Glossary

Term Meaning Example
Variable A letter representing an unknown value x, y, a, b
Constant A fixed number that does not change 3, −7, 0, 100
Term A single constant, variable, or product of constants and variables 3x, −5y², 7, 2xy
Coefficient The numerical part of a term In 5x², coefficient = 5; in -y, coefficient = -1
Expression Combination of terms connected by + or − 3x + 2y − 7
Like Terms Terms with the SAME variable(s) raised to SAME powers 3x² and −5x² are like terms. 3x and 3x² are NOT.
Unlike Terms Terms with different variables or different powers 3x and 5y; 2x² and 2x

📝 2. Types of Algebraic Expressions

Type Number of Terms Example
Monomial 1 5x², −3y, 7
Binomial 2 x + 3, 2a − 5b
Trinomial 3 x² + 3x − 4, a + b + c
Polynomial Any number of terms All of the above are polynomials

➕ 3. Addition and Subtraction

Rule: Only like terms can be added or subtracted. Combine their coefficients; variable part stays same.

Example 1 — Add: (3x² + 5x − 7) + (2x² − 3x + 4)
= (3+2)x² + (5−3)x + (−7+4) = 5x² + 2x − 3

Example 2 — Subtract: (5a² − 4a + 3) − (2a² + 3a − 7)
= 5a² − 4a + 3 − 2a² − 3a + 7 (distribute the minus sign)
= (5−2)a² + (−4−3)a + (3+7) = 3a² − 7a + 10
⚠️ Sign Change Rule When subtracting an expression, change the sign of EVERY term inside the bracket:
−(3x − 2y + 5) = −3x + 2y − 5 (every sign flips!)

✖️ 4. Multiplication of Algebraic Expressions

Case Rule Example
Monomial × Monomial Multiply coefficients; add exponents of same variable 3x² × 4x³ = 12x⁵
Monomial × Polynomial Distribute monomial to each term (distributive law) 2x(3x + 4y) = 6x² + 8xy
Binomial × Binomial Multiply each term of first with each term of second (FOIL) (x+3)(x+4) = x²+4x+3x+12 = x²+7x+12
General Polynomial × Polynomial Each term of first multiplied by each term of second, then combine like terms (x+2)(x²+3x−1) = x³+3x²−x+2x²+6x−2 = x³+5x²+5x−2

🏠 5. Important Algebraic Identities

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
Practical use of identities (smart calculation!):
• 98² = (100−2)² = 10000 − 400 + 4 = 9604
• 103² = (100+3)² = 10000 + 600 + 9 = 10609
• 99 × 101 = (100−1)(100+1) = 10000 − 1 = 9999

📊 6. Value of an Expression

To find the value of an expression, substitute (replace) the variable with the given number and simplify.

Find value of 3x² − 5x + 2 at x = 3:
= 3(3)² − 5(3) + 2 = 3×9 − 15 + 2 = 27 − 15 + 2 = 14
✏️ Practice Questions — Algebraic Expressions (30 Questions)

Section A — Terms and Types Easy

Q1. State the number of terms in: 4xy + 3x²y − 7 + z
Q2. Identify like terms in: 3x, -5y, 7x, 2y², -4x, 8y
Q3. Write coefficient of x² in −6x² + 3x − 2
Q4. Classify: 7x (monomial/binomial/trinomial)
Q5. Find value of 2x − 5 when x = 4
Q6. Add: (3a + 5b) + (2a − 3b)
Q7. Subtract: (5x − 3y) − (2x + y)
Q8. Simplify: 4m + 3n − 2m + 5n − n

Section B — Operations Medium

Q9. Add: (3x² − 4x + 1) + (2x² + 5x − 3)
Q10. Subtract: (7a² − 3a + 5) − (4a² + 2a − 1)
Q11. Multiply: 3x × (2x − 5y)
Q12. Multiply: (x + 4)(x + 7)
Q13. Multiply: (2a − 3b)(4a + 5b)
Q14. Find value of 5x² − 3x + 1 at x = −2
Q15. Find value of 2a² − 3b + 4c at a=1, b=2, c=3
Q16. Simplify: (3x + 2)² using identity

Section C — Identities Medium

Q17. Expand: (4x + 5y)²
Q18. Expand: (3a − 2b)²
Q19. Evaluate using identity: 105²
Q20. Evaluate: 97²
Q21. Find: 105 × 95 using identity (a+b)(a-b)=a²-b²
Q22. Find: 1003 × 997

Section D — Challenge Hard

  1. Q23. What must be added to 3x² − 4x + 1 to get 5x² + 2x − 3?
  2. Q24. What must be subtracted from 7a − 3b to get 2a + b?
  3. Q25. If P = 3x² − 2x + 5 and Q = 2x² + x − 3, find: (i) P + Q (ii) P − Q (iii) Value of P + Q at x = 1
  4. Q26. Multiply: (x + 2)(x² − 2x + 4). What pattern do you notice? Express as a³ + b³.
  5. Q27. If a + b = 7 and ab = 12, find a² + b² using identity.
  6. Q28. If x − y = 5 and xy = 24, find x² + y².
  7. Q29. The perimeter of a rectangle is (16a − 4b) units. If length = (5a + b), find width and area.
  8. Q30. If x + 1/x = 5, find x² + 1/x² using (x + 1/x)² = x² + 2 + 1/x².
✅ Key Answers: Q1:4 terms | Q3:-6 | Q5:3 | Q6:5a+2b | Q9:5x²+x-2 | Q10:3a²-5a+6 | Q11:6x²-15xy | Q12:x²+11x+28 | Q14:27 | Q16:9x²+12x+4 | Q19:11025 | Q20:9409 | Q21:9975 | Q22:999991 | Q23:2x²+6x-4 | Q27:a²+b²=25 | Q28:x²+y²=73 | Q30:x²+1/x²=23

📝 Quick Revision

  1. Terms connected by + or − form an expression. Coefficients are the numerical factors.
  2. Like terms: same variable, same power. Only like terms can be added/subtracted.
  3. When subtracting, change sign of EVERY term in the bracket.
  4. Multiplying: Coefficients multiply, exponents add (aᵐ × aⁿ = aᵐ⁺ⁿ).
  5. (a+b)² = a²+2ab+b² | (a-b)² = a²-2ab+b² | (a+b)(a-b) = a²-b²
  6. To find value: substitute value and simplify using BODMAS.