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9.Verify :
i)\(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
ii)\(x^3 - y^3 = (x - y)(x^2 + xy + y^2) \)
Answer :

i)We know that,
\((x + y)^3 = x^3 + y^3 + 3xy(x + y)\)
\(x^3 + y^3 = (x + y)^3 - 3xy(x + y)\)
\(= (x + y)[(x + y)^2 - 3xy]\)
\(= (x + y)[(x^2 + y^2 +2xy - 3xy]\)
\(= (x + y)[(x^2 + y^2 - xy]\)
\(= R.H.S\)
Hence, Proved.

ii)We know that,
\((x - y)^3 = x^3 - y^3 - 3xy(x - y)\)
\(x^3 - y^3 = (x - y)^3 + 3xy(x - y)\)
\(= (x - y)[(x - y)^2 + 3xy]\)
\(= (x - y)[(x^2 + y^2 - 2xy + 3xy]\)
\(= (x - y)[(x^2 + y^2 + xy]\)
\(= R.H.S\)
Hence, Proved.