Question

verify x cube - y cube=(x-y) (x sqaure+xy+y sqaure)

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Answer 1

Answer:o 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.

Step-by-step explanation:

Answer 2

Answer:

${x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2} )$

$let \: x = 3 \: and \: y = 2$

${3}^{3} - {2}^{3} = (3 - 2)( {3}^{2} + 3 \times 2 + {2}^{2} )$

$(27 - 8) = 1 \times (9 + 6 + 4)$

$19 = 19$

$lhs = rhs$

verify it