Introduction to Remarkable Identities

Introduction to Remarkable Identities

• Remarkable identities are specific algebraic formulas that simplify expressions and calculations.
• They are frequently used in algebra to factorize expressions, expand products, and solve equations.
• Understanding these identities is essential for success in algebra.
• They represent special products of binomials
• Memorizing these identities can save significant time during mathematical calculations.

Key Identities

• Square of a sum: (a + b)² = a² + 2ab + b²
  • This formula expands expressions of the form (a + b) squared.
  • Example: (x + 3)² = x² + 6x + 9
  • Example: (2x + 5y)² = 4x² + 20xy + 25y²
• Square of a difference: (a – b)² = a² – 2ab + b²
  • This identity expands expressions like (a - b) squared.
  • Example: (y – 5)² = y² – 10y + 25
  • Example: (3a - 4b)² = 9a² - 24ab + 16b²
• Difference of two squares: a² – b² = (a + b)(a – b)
  • This identity factors expressions with the squared difference of two terms.
  • Example: x² – 9 = (x + 3)(x – 3) -Example: 4x² - 49 = (2x + 7 )(2x-7)
• Cube of a sum: (a + b)³ = a³ + 3a²b + 3ab² + b³
  • This formula expands expressions like (a + b) cubed.
  • Example: (x + 2)³ = x³ + 6x² + 12x + 8 -Example:(y+2z)³ = y³+6y²z+12yz²+8z³
• Cube of a difference: (a – b)³ = a³ – 3a²b + 3ab² – b³
  • This identity expands expressions of the form (a – b) cubed.
  • Example: (y – 1)³ = y³ – 3y² + 3y – 1
  • Example: (p - 1)³ = p³ - 3p² + 3p - 1

Applications and Examples

• Factoring:
  • Use applicable identities to simplify expressions and factor them.
  • Example: x² + 8x + 16 = (x + 4)²
• Expanding expressions:
  • Apply the identities to expand and simplify algebraic expressions.
  • Example: (2x – 3y)² = 4x² – 12xy + 9y²
• Solving equations:
  • Solve equations using identities, factoring, and isolating variables.
  • Example: If a² – 49 = 0, then (a + 7)(a – 7) = 0, so a = 7 or a = -7.
• Simplifying expressions:
• Factoring expressions: Decomposing expressions into factors to solve equations or evaluate algebraic expressions.
• Solving equations: Finding the values for unknown variables to make an equation true.
• Proving geometric theorems: Using algebraic manipulations to verify geometric claims.

Additional Notes

• Memorizing and understanding the identities is crucial.
• Applying the identities correctly is essential for problem-solving.
• Numerous numerical examples demonstrate the applications of these identities.
• Grade 8 focuses on straightforward applications of these identities, without excessive complexity.