Class 8 Notes

Class 8 Mathematics Notes Chapter 9 (Algebraic Expressions and Identities) – Mathematics Book

Philoid

Philoid

— 5 min read

Mathematics
Alright class, let's focus on Chapter 9: Algebraic Expressions and Identities. This is a foundational chapter, and understanding it well is crucial not just for your Class 8 exams but also for various government exams where quantitative aptitude is tested. Pay close attention!

Chapter 9: Algebraic Expressions and Identities - Detailed Notes

1. What are Expressions?

  • An expression is a combination of constants, variables, and mathematical operators (+, -, ×, ÷).
  • Example: 5x, 2x - 3, 4xy + 7, x² + 1

2. Terms, Factors, and Coefficients

  • Terms: Parts of an expression separated by '+' or '-' signs.
    • In 4xy + 7, the terms are 4xy and 7.
  • Factors: When a term is formed as a product of numbers and/or variables, these numbers/variables are its factors.
    • In the term 4xy, the factors are 4, x, and y.
    • The term 7 has only one factor, 7.
  • Coefficient: The numerical factor of a term is called its coefficient.
    • In 4xy, the coefficient is 4.
    • In -5x²y, the coefficient is -5.
    • If a term is just a variable like y, the coefficient is 1. If it's -y, the coefficient is -1.

3. Monomials, Binomials, Trinomials, and Polynomials

  • Monomial: An algebraic expression containing only one term.
    • Examples: 7xy, -5m, 3z², 4
  • Binomial: An algebraic expression containing two unlike terms.
    • Examples: x + y, m - 5, a² + b², 2x + 3y
  • Trinomial: An algebraic expression containing three unlike terms.
    • Examples: x + y + 7, ab + a + b, 2x² - 3x + 5
  • Polynomial: An algebraic expression containing one or more terms with non-negative integer exponents for the variables. Monomials, binomials, and trinomials are all types of polynomials.
    • An expression like x + 1/x or √y + 3 is not a polynomial because the powers of variables are not non-negative integers (1/x = x⁻¹, √y = y¹/²).

4. Like and Unlike Terms

  • Like Terms: Terms that have the same algebraic factors (same variables raised to the same power). The coefficients can be different.
    • Examples: 7x and -13x; 5x²y and 9x²y; -3pq and pq
  • Unlike Terms: Terms that have different algebraic factors.
    • Examples: 7x and 7y; 5x²y and 9xy²; -3pq and -3p
  • Importance: Addition and subtraction operations can only be performed between like terms.

5. Addition and Subtraction of Algebraic Expressions

  • Rule: Combine only the like terms. Add or subtract their coefficients, keeping the algebraic factors the same. Unlike terms remain as they are.
  • Method 1: Horizontal Method
    • Write expressions in a row. Rearrange terms to group like terms together. Combine like terms.
    • Example (Addition): Add (7xy + 5yz - 3zx) and (4yz + 9zx - 4y)
      = 7xy + 5yz - 3zx + 4yz + 9zx - 4y
      = 7xy + (5yz + 4yz) + (-3zx + 9zx) - 4y
      = 7xy + 9yz + 6zx - 4y
    • Example (Subtraction): Subtract (5x² - 4y² + 6y - 3) from (7x² - 4xy + 8y² + 5x - 3y)
      = (7x² - 4xy + 8y² + 5x - 3y) - (5x² - 4y² + 6y - 3)
      = 7x² - 4xy + 8y² + 5x - 3y - 5x² + 4y² - 6y + 3 (Change signs of the subtracted expression)
      = (7x² - 5x²) - 4xy + (8y² + 4y²) + 5x + (-3y - 6y) + 3
      = 2x² - 4xy + 12y² + 5x - 9y + 3
  • Method 2: Column Method
    • Write expressions one below the other such that like terms are in the same column. Add or subtract column-wise. Remember to change signs of the lower expression during subtraction.

6. Multiplication of Algebraic Expressions

  • Rule 1 (Signs): (+) × (+) = (+); (-) × (-) = (+); (+) × (-) = (-); (-) × (+) = (-)

  • Rule 2 (Exponents): When multiplying terms with the same base variable, add their exponents (Law of Exponents: aᵐ × aⁿ = aᵐ⁺ⁿ).

  • a) Multiplying a Monomial by a Monomial:

    • Multiply the coefficients.
    • Multiply the variable parts (using the law of exponents).
    • Example: (3x) × (5xy) = (3 × 5) × (x × x × y) = 15x¹⁺¹y = 15x²y
    • Example: (-4ab) × (-2a²bc) = (-4 × -2) × (a × a²) × (b × b) × c = 8a¹⁺²b¹⁺¹c = 8a³b²c

  • b) Multiplying a Monomial by a Polynomial:

    • Use the Distributive Property: Multiply the monomial by each term of the polynomial.
    • Example: 3p × (4p² + 5p + 7) = (3p × 4p²) + (3p × 5p) + (3p × 7)
      = 12p¹⁺² + 15p¹⁺¹ + 21p
      = 12p³ + 15p² + 21p

  • c) Multiplying a Polynomial by a Polynomial:

    • Multiply each term of the first polynomial by each term of the second polynomial. Group like terms and combine them.
    • Example: (x + 7) × (x + 2)
      = x × (x + 2) + 7 × (x + 2) (Distributive property)
      = (x × x) + (x × 2) + (7 × x) + (7 × 2) (Distributive property again)
      = x² + 2x + 7x + 14
      = x² + 9x + 14 (Combine like terms)
    • Example: (a - b) × (2a + 3b - c)
      = a × (2a + 3b - c) - b × (2a + 3b - c)
      = (a × 2a) + (a × 3b) + (a × -c) + (-b × 2a) + (-b × 3b) + (-b × -c)
      = 2a² + 3ab - ac - 2ab - 3b² + bc
      = 2a² + (3ab - 2ab) - ac - 3b² + bc
      = 2a² + ab - ac - 3b² + bc

7. What is an Identity?

  • An identity is an equality that holds true for all possible values of the variables involved.
  • Example: (a + 1)(a + 2) = a² + 3a + 2. If you put a = 1, LHS = (2)(3) = 6, RHS = 1+3+2 = 6. If you put a = -1, LHS = (0)(1) = 0, RHS = 1-3+2 = 0. It's true for any value of 'a'.
  • This is different from an equation, which is true only for specific values of the variable(s). Example: x + 5 = 8 is true only for x = 3.

8. Standard Identities
These are very important and frequently used for simplification and factorization. Memorize them!

  • Identity I: (a + b)² = a² + 2ab + b²

    • Example: (2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y²
    • Example (Calculation): 103² = (100 + 3)² = 100² + 2(100)(3) + 3² = 10000 + 600 + 9 = 10609

  • Identity II: (a - b)² = a² - 2ab + b²

    • Example: (4p - 3q)² = (4p)² - 2(4p)(3q) + (3q)² = 16p² - 24pq + 9q²
    • Example (Calculation): 98² = (100 - 2)² = 100² - 2(100)(2) + 2² = 10000 - 400 + 4 = 9604

  • Identity III: (a + b)(a - b) = a² - b²

    • Example: (5m + 3n)(5m - 3n) = (5m)² - (3n)² = 25m² - 9n²
    • Example (Calculation): 51 × 49 = (50 + 1)(50 - 1) = 50² - 1² = 2500 - 1 = 2499

  • Identity IV: (x + a)(x + b) = x² + (a + b)x + ab

    • Example: (y + 3)(y + 5) = y² + (3 + 5)y + (3 × 5) = y² + 8y + 15
    • Example: (2t + 5)(2t - 3) = (2t)² + (5 + (-3))(2t) + (5 × -3) = 4t² + (2)(2t) - 15 = 4t² + 4t - 15 (Here x = 2t, a = 5, b = -3)
    • Example (Calculation): 103 × 104 = (100 + 3)(100 + 4) = 100² + (3 + 4)100 + (3 × 4) = 10000 + 700 + 12 = 10712

Key Takeaways for Exams:

  • Be very clear about the difference between terms, factors, and coefficients.
  • Master identifying like and unlike terms for addition/subtraction.
  • Practice multiplication carefully, especially polynomial by polynomial, paying attention to signs and exponents.
  • Memorize the four standard identities and practice applying them for both expanding expressions and simplifying numerical calculations quickly.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on this chapter for your practice:

  1. The coefficient of y² in the expression -5x²y² + 7xy - 3 is:
    A) -5
    B) -5x²
    C) 5x²
    D) 7x

  2. Which of the following is a binomial?
    A) 7xy + 5x - 3y
    B) 4a²b
    C) p² - q²
    D) 100

  3. The sum of (ab - bc) and (bc - ca) is:
    A) ab - ca
    B) ab + ca
    C) 2bc
    D) ab - 2bc - ca

  4. On subtracting (5a² - 3b² + 2ab) from (a² + b² - ab), we get:
    A) -4a² + 4b² - 3ab
    B) 4a² - 4b² + 3ab
    C) 6a² - 2b² + ab
    D) -4a² - 4b² - 3ab

  5. The product of (-6p²q) and (3pq²r) is:
    A) 18p³q³r
    B) -18p²q²r
    C) -18p³q³r
    D) -3p³q³r

  6. The result of 2x (3x + 5y) is:
    A) 6x² + 5y
    B) 6x + 10xy
    C) 6x² + 10xy
    D) 5x² + 7xy

  7. The product (x + 5)(x - 3) is equal to:
    A) x² + 8x - 15
    B) x² - 2x - 15
    C) x² + 2x + 15
    D) x² + 2x - 15

  8. Using a suitable identity, the value of (102)² is:
    A) 10404
    B) 10004
    C) 10400
    D) 10204

  9. Using the identity (a - b)² = a² - 2ab + b², the value of (9.9)² is:
    A) 98.01
    B) 99.81
    C) 98.1
    D) 9.801

  10. The expression (3x + 7)² - 84x simplifies to:
    A) (3x - 7)²
    B) (9x + 49)²
    C) (3x + 7)²
    D) (9x - 49)²

Answer Key:

  1. B
  2. C
  3. A
  4. A (Remember to change signs when subtracting)
  5. C
  6. C
  7. D (Using (x+a)(x+b) = x² + (a+b)x + ab, where a=5, b=-3)
  8. A (Using (100+2)² = 100² + 21002 + 2²)
  9. A (Using (10 - 0.1)² = 10² - 2100.1 + (0.1)²)
  10. A (Expand (3x+7)² = 9x² + 42x + 49. Then 9x² + 42x + 49 - 84x = 9x² - 42x + 49, which is (3x-7)²)

Study these notes thoroughly and practice more problems. Good luck with your preparation!

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