Identifying Special Situations in Factoring

• Difference of two squares
  • a²- b² = (a + b)(a - b)
    • a²-81 ^{2 = (a + 9)( a- 9)}
    • a²- 4^{2 =( a + 2)(a - 2)}
    • a²- 36^{2 = ( a + 6)(a-6)} 
• Trinomial perfect squares
  • a² + 2ab + b²= (a + b)(a + b) or (a + b)²
    • a² + 6a + 9²= (a + 3)(a + 3) or (a + 3)²
    • ^{a2 + 14a + 492= (a + 7)(a + 7) or (a + 7)2}
    • ^{a2 + 18a + 812= (a + 9)(a + 9) or (a + 9)2}
  • ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
    • a² - 8ab + 16² = (a - 4)(a - 4) or (a - 4)²
    • ^{a2 - 6a + 92 = (a - 3)(a - 3) or (a - 3)2}
    • ^{a2 - 12a + 362 = (a - 6)(a - 6) or (a - 6)2}
• Difference of two cubes
  • a³ - b³
    • 3 - cube root 'em
    • 2 - square 'em
    • 1 - multiply and change
      • a³ - 1 ^{= (a-1)(a+a+1)}
      • ^{a3 - 27 = (a-3)(a-3a+9)} 
      • ^{a3 - 125 =(a-5)(a+5a+25)}
• Sum of two cubes
  • a³ + b³ 
    • 3 - cube root 'em
    • 2 - square 'em
    • 1 - multiply and change
      • a³ + 1 = (a + 1)(a²+a+1)
      • ^{a3 + 125 =} (a+5)(a²+5a+25)
      • ^{a3 + 64 =}  (a+4)(a²+4a+16)
• Binomial expansion
  • (a + b)³ = a³ + 3a²b + 3ab² + b³= (a + 4) ³= a³ + 12a² + 48a + 64
  • (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ =(a + 3) ⁴  = a⁴  + 12a³ + 54a² + 108a + 81