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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.

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Student-friendly 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.

How to write this in exams

  1. 1

    Start with the exact idea

    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.

  2. 2

    Then show how to use it

    Name the three terms as a, b, c. Write the identity fully. Substitute each term carefully and simplify like terms.

  3. 3

    Add one concrete example

    (x+2+3)^2 = x^2+4+9+4x+12+6x = x^2+10x+25.

  4. 4

    Avoid this incomplete answer

    Writing (a+b+c)^2=a^2+b^2+c^2+2ab is wrong because it misses 2bc and 2ca.

Definition

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.

Example

(x+2+3)^2 = x^2+4+9+4x+12+6x = x^2+10x+25.

Rule to remember

(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca; a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca).

Memory hook

Three terms give three squares and three doubled pairs.

Examples and method

Worked example

Expand (x+y+2)^2. It becomes x^2+y^2+4+2xy+4y+4x after writing all squares and doubled pair products.

Method to apply

Name the three terms as a, b, c. Write the identity fully. Substitute each term carefully and simplify like terms.

How CBSE asks it

Questions may ask expansion, simplification using a+b+c=0, or proving a cube relation under a given condition.

Avoid common mistakes

Common confusion

Students often forget one of the double-product terms, especially 2ca.

Common wrong answer

Writing (a+b+c)^2=a^2+b^2+c^2+2ab is wrong because it misses 2bc and 2ca.

Exam tip

For (a+b+c)^2, write all three squares first, then all three pair products doubled.

Quick check

When can we directly write a^3+b^3+c^3=3abc?

We can write a^3+b^3+c^3=3abc only when a+b+c=0. Without this condition, the equality is not generally true.

Answer writing and exam use

1-mark answer

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.

2-mark answer

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. (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca; a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca). (x+2+3)^2 = x^2+4+9+4x+12+6x = x^2+10x+25.

3-mark answer

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. (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca; a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca). Expand (x+y+2)^2. It becomes x^2+y^2+4+2xy+4y+4x after writing all squares and doubled pair products. Questions may ask expansion, simplification using a+b+c=0, or proving a cube relation under a given condition. Writing (a+b+c)^2=a^2+b^2+c^2+2ab is wrong because it misses 2bc and 2ca.
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