Algebraic Identities List with Examples and Extra Questions

All the standard Algebraic Identities are gotten from the Binomial Theorem, which is given as:

Knowing these standard logarithmic personalities will help you in doing snappy math and spare time in focused exams, which is vital. Following is the arithmetical characters list that you should know:

Identity I: (a + b)2 = a2 + 2ab + b2
Identity II: (a – b)2 = a2 – 2ab + b2
Identity III: a2 – b2= (a + b)(a – b)
Identity IV: (x + a)(x + b) = x2 + (a + b)x + ab
Identity V: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Identity VI: (a + b)3 = a3 + b3 + 3ab (a + b)
Identity VII: (a – b)3 = a3 – b3 – 3ab (a – b)
Identity VIII: a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

Examples:

Example 1: Find the product of (x + 1)(x + 1) using standard algebraic identities.
- Solution: (x + 1)(x + 1) can be written as (x + 1)2. Thus, it is of the form Identity I where a = x and b = 1. So we have,
  (x + 1)2 = (x)2 + 2(x)(1) + (1)2 = x2 + 2x + 1

Example 2: Factorise (x4 – 1) using standard algebraic identities.
- Solution: (x4 – 1) is of the form Identity III where a = x2 and b = 1. So we have,
  (x4 – 1) = ((x2)2– 12) = (x2 + 1)(x2 – 1)
  
  The factor (x2 – 1) can be further factorised using the same Identity III where a = x and b = 1. So,
  (x4 – 1) = (x2 + 1)((x)2 –(1)2) = (x2 + 1)(x + 1)(x – 1)

Practice Questions:

Question 1: Expand (3x – 4y)3 using standard algebraic identities.
Question 2: Factorize (x3 + 8y3 + 27z3 – 18xyz) using standard algebraic identities.
Question 3: If a + b + c = 6 , a2 + b2 + c2 = 14 and ab + bc + ca = 11 , what is a3 + b3 + c3 – 3abc ?

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