Three-Variable Identities
Three-variable identities expand or factorise expressions containing a, b, and c together, especially (a+b+c)2 and a3+b3+c3-3abc.
In one line: Three terms give three squares and three doubled pairs.
Practice This ConceptUnderstand it
The identity (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca includes three square terms and three double-product terms. Another important identity is a3+b3+c3-3abc = (a+b+c)(a2+b2+c2-ab-bc-ca), and if a+b+c=0, then a3+b3+c3=3abc.
Example
Formula and rule
(a+b+c)2=a2+b2+c2+2ab+2bc+2ca; a3+b3+c3-3abc=(a+b+c)(a2+b2+c2-ab-bc-ca).
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
- 1Expand (x+y+2)2.
- 2It becomes x2+y2+4+2xy+4y+4x after writing all squares and doubled pair products.
Method you can reuse
- 1Name the three terms as a, b, c.
- 2Write the identity fully.
- 3Substitute each term carefully and simplify like terms.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
When can we directly write a3+b3+c3=3abc?
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Common wrong answer
Writing (a+b+c)2=a2+b2+c2+2ab.
Why it loses marks: It misses 2bc and 2ca.
Don't mix this up
Exam tip
Write it in the exam
Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.
How CBSE asks it
- 1
Enough for 1 mark
Three-variable identities expand or factorise expressions containing a, b, and c together, especially (a+b+c)2 and a3+b3+c3-3abc.
- 2
Add for 2 marks
Example: (x+2+3)2 = x2+4+9+4x+12+6x = x2+10x+25.
- 3
Add for 3 marks
The identity (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca includes three square terms and three double-product terms.
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