Class 8 Notes - Algebraic Expressions and Identities

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Algebraic expressions and identities form the foundation of algebra. This chapter focuses on understanding algebraic expressions, operations on them, and standard identities that simplify algebraic manipulations.

Algebraic Expressions

Definition

An algebraic expression is a combination of constants, variables, and arithmetic operations (+, −, ×, ÷). Examples include:

  • 5x + 7
  • 3x² − 2y + z
  • 2(a + b) − 3c

Terms of an Expression

An expression is made up of terms separated by '+' or '−' signs. For example:

  • In 5x + 7, the terms are 5x and 7.
  • In 3x² − 2y + z, the terms are 3x², −2y, and z.

Types of Terms

  • Like Terms: Terms having the same variables raised to the same powers. Example: 3x and −5x.
  • Unlike Terms: Terms having different variables or powers. Example: 3x and 2y.

Types of Algebraic Expressions

  • Monomial: An expression with a single term. Example: 7x.
  • Binomial: An expression with two terms. Example: 3x + 5.
  • Trinomial: An expression with three terms. Example: x² + 2x + 3.
  • Polynomial: An expression with one or more terms. Example: x³ + 2x² + 3x + 5.

Operations on Algebraic Expressions

Addition and Subtraction

To add or subtract algebraic expressions, combine like terms.

Example:

Add: (3x + 5y) + (2x − 3y)

Solution: (3x + 2x) + (5y − 3y) = 5x + 2y

Multiplication

Multiplication involves multiplying each term of one expression with every term of the other.

Example:

Multiply: (x + 2)(x + 3)

Solution: x² + 3x + 2x + 6 = x² + 5x + 6

Division

In division, each term of the numerator is divided by the denominator, if possible.

Example:

Divide: (4x² + 6x) ÷ 2x

Solution: 2x + 3

Algebraic Identities

Definition

Algebraic identities are standard results that hold true for all values of the variables involved. They simplify complex algebraic calculations.

Standard Identities

  • (a + b)² = a² + 2ab + b²
  • (a − b)² = a² − 2ab + b²
  • (a + b)(a − b) = a² − b²
  • (x + a)(x + b) = x² + (a + b)x + ab

Example:

Simplify: (x + 3)(x + 4)

Solution: Using (x + a)(x + b) = x² + (a + b)x + ab, we get x² + 7x + 12.

Applications of Identities

  • Expanding expressions
  • Simplifying calculations
  • Finding squares and products quickly

Practice Problems

  1. Simplify: (2x + 3)(2x − 3)
  2. Expand: (a + b + c)²
  3. Find the product: (x + 4)(x − 5)
  4. Verify the identity: (a + b)² = a² + 2ab + b² for a = 2, b = 3.

Understanding and practicing these concepts is essential for mastering algebra. These foundational skills are crucial for advanced mathematics and real-world problem-solving.