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Algebraic Identities

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An algebraic identity is an algebraic equation that is true for all values of the variables occurring in it. 
Identity I : `(x + y)^2 = x^2 + 2xy + y^2 `

Identity II :
`(x – y)^2 = x^2 – 2xy + y^2` 

Identity III :
`x^2 – y^2 = (x + y) (x – y)` 

Identity IV :
`(x + a) (x + b) = x^2 + (a + b)x + ab`

Identity V :
 `(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx`

Remark :
 We call the right hand side expression the expanded form of the left hand side expression. Note that the expansion of `(x + y + z)^2` consists of three square terms and three product terms.

Identity VI :` (x + y)^3 = x^3 + y^3 + 3xy (x + y)`

Identity VII :
 `(x – y)^3 = x^3 – y^3 – 3xy(x – y) `
= `x^3 – 3x^2y + 3xy^2 – y^3`

Identity VIII : `x^3 + y^3 + z^3 – 3xyz = (x + y + z)(x^2 + y^2 + z^2 – xy – yz – zx)`

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( a + b )2 = a2 + 2ab + b2 .

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