Algebraic Identities Lecture Notes

Introduction to Algebraic Identities

• Definition: An algebraic identity is an equality valid for all values of the variables within it.
• Equation vs Identity: An equation is only true for specific values of its variables, whereas an identity holds universally.

Standard Algebraic Identities

Identity I: ((a+b)^2 = a^2 + 2ab + b^2)

• Derivation:
  • ((a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2)

Identity II: ((a-b)^2 = a^2 - 2ab + b^2)

• Derivation:
  • ((a-b)^2 = (a-b)(a-b) = a(a-b) - b(a-b) = a^2 - ab - ba + b^2 = a^2 - 2ab + b^2)

Identity III: ((a+b)(a-b) = a^2 - b^2)

• Derivation:
  • ((a+b)(a-b) = a(a-b) + b(a-b) = a^2 - ab + ba - b^2 = a^2 - b^2)

Identity IV: ((x+a)(x+b) = x^2 + (a+b)x + ab)

• Derivation:
  • ((x+a)(x+b) = x(x+b) + a(x+b) = x^2 + xb + ax + ab = x^2 + (a+b)x + ab)

Special Cases of Standard Identities

• For (b = a): ((x+a)^2 = x^2 + 2ax + a^2)
• For (b = -c, a = -c): ((x-c)^2 = x^2 - 2cx + c^2)
• Other similar transformations are noted.

Other Algebraic Identities

Identity V: ((x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx)

Identity VI: ((x+y)^3 = x^3 + y^3 + 3xy(x+y))

Identity VII: ((x-y)^3 = x^3 - y^3 - 3xy(x-y))

Identity VIII: (x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - xy - yz - zx))

Identity IX: ((a+b+c)^3 = a^3 + b^3 + c^3 + 3a^2b + 3a^2c + 3b^2c + 3b^2a + 3c^2a + 3c^2b + 6abc)

Solved Examples

• Example 1: Expand ((3a+4b+c)^2) using identity V.
• Example 2: Factorize (27x^3 + y^3 + 27z^3 - 27xyz) using identity VIII.

Summary of All Algebraic Identities

• A table or visual summarizing the identities could be useful.

Additional Resources

• Links to further readings and related topics like BODMAS rule, surds in maths, etc.

These notes provide a comprehensive overview of algebraic identities, their derivations, and useful applications in mathematical problems.