Three-Variable Identities
Three-variable identities expand or factorise expressions containing a, b, and c together, especially (a+b+c)^2 and a^3+b^3+c^3-3abc.
Practice This ConceptMain explanation
Teacher explanation
The identity (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca includes three square terms and three double-product terms. Another important identity is a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca), and if a+b+c=0, then a^3+b^3+c^3=3abc.
Example
(x+2+3)^2 = x^2+4+9+4x+12+6x = x^2+10x+25.
Simple analogy
Three terms give three squares and three doubled pairs.
Common confusion
Students often forget one of the double-product terms, especially 2ca.
Exam tip
For (a+b+c)^2, write all three squares first, then all three pair products doubled.
Answer writing and exam use
1-mark use
Write the exact meaning of three-variable identities in one clean line.
2-mark use
Define three-variable identities and add one example or condition.
3-mark use
Explain three-variable identities, show the method or example, and mention the common mistake.
Practice this concept with focused MCQs
Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.
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