Prove that (X+Y)^3-(X-Y)^3-6Y(X^2-Y^2)=8Y^3

Asked by abinash.gupta003 | 21st Jul, 2016, 09:04: PM

Expert Answer:

begin mathsize 16px style Consider comma space LHS space equals space left parenthesis straight X plus straight Y right parenthesis cubed space minus space left parenthesis straight X minus straight Y right parenthesis cubed space minus space 6 straight Y left parenthesis straight X squared minus straight Y squared right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight X cubed plus straight Y cubed space plus thin space 3 straight X squared straight Y space plus thin space 3 XY squared space minus space left parenthesis straight X cubed minus straight Y cubed space minus thin space 3 straight X squared straight Y space plus thin space 3 XY squared right parenthesis minus space 6 straight X squared straight Y space plus space 6 straight Y cubed
space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight X cubed plus straight Y cubed space plus thin space 3 straight X squared straight Y space plus thin space 3 XY squared space minus space straight X cubed plus straight Y cubed space plus thin space 3 straight X squared space minus thin space 3 XY squared minus space 6 straight X squared straight Y space plus space 6 straight Y cubed
space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight Y cubed space plus straight Y cubed plus space 6 straight Y cubed
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 8 straight Y cubed
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space RHS
Hence space proved. end style

Answered by Rebecca Fernandes | 21st Jul, 2016, 10:38: PM

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