Concept: Algebraic Formulae


(a + b + ...)2

  • (a)2 = a2
  • (a + b)2 = a2 + b2 + 2ab
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (a + b + c + d)2 = a2 + b2 + c2 + d2 + 2ab + 2ac + 2ad + 2bc + 2bd + 2 cd

 

(a + b)n

  • (a + b)1 = a1 + b1
  • (a + b)2 = a2 + 2ab + b2
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3
  • (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
  • (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5

Remember:
  • (a + b)n = nC0anb0nC1an-1b2 + nC2an-2b2 + ... + nCn-2a2bn-2 + nCn-3a1bn-1nCna0bn​​​​​​​
    [This is called Binomial Theorem]
  • Number of terms in (a + b)n = (n + 1)

 

(a - b)n

  • (a - b)1 = a1 - b1
  • (a - b)2 = a2 - 2ab + b2
  • (a - b)3 = a3 - 3a2b + 3ab2 - b3
  • (a - b)4 = a4 - 4a3b + 6a2b2 - 4ab3 + b4
  • (a - b)5 = a5 - 5a4b + 10a3b2 - 10a2b3 + 5ab4 - b5

 

an - bn

  • a1 - b1 = a - b
  • a2 - b2 = (a - b)(a + b)
  • a3 - b3 = (a - b)(a2 + ab + b2)
  • a4 - b4 = (a - b)(a3 + a2b + ab2 + b3) = (a  - b)(a + b)(a2 + b2)
  • a5 - b5 = (a - b)(a4 + a3b + a2b2 + ab3 + b4)

Remember:
  • an - bn = (an-1 + an-2b + ... + abn-2 + bn-1)
  • an - bn is always divisible by (a - b)
  • an - bn is also divisible by (a + b) if n is even.

 

an + bn

  • a1 + b1 = a + b
  • a2 + b2 = No Identity
  • a3 + b3 = (a + b)(a2 - ab + b2)
  • a4 + b4 = No Identity
  • a5 + b5 = (a + b)(a4 - a3b + a2b2 - ab3 + b4)
Remember:
  • an + bn is divisible by (a + b) if n is odd.

 

a3 + b3 + c3

a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

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