013: Verify that x + y + 1} – 3xy1 =} (x + y + 2)[(x - y)2 + (y - 3)2 + (2 - x)”]
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ANSWER
To prove: x
3
−y
3
=(x−y)(x
2
+xy+y
2
)
Consider the right hand side (RHS) and expand it as follows:
(x−y)(x
2
+xy+y
2
)=x
3
+x
2
y+xy
2
−yx
2
−xy
2
−y
3
=(x
3
−y
3
)+(x
2
y+xy
2
+x
2
y−xy
2
)=x
3
−y
3
=LHS
Hence proved.
Yes, we can call it as an identity: For example:
Let us take x=2 and y=1 in x
3
−y
3
=(x−y)(x
2
+xy+y
2
) then the LHS and RHS will be equal as shown below:
2
3
−1
3
=7 and
(2−1)(2
2
+(2×1)+1
2
)=1(5+2)=1×7=7
Therefore, LHS=RHS
Hence, x
3
−y
3
=(x−y)(x
2
+xy+y
2
) can be used as an identity.
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