Question

If A = x2+ xy + y2 and B = x - y, prove that, AB = x3 - y3.

Answer 1

Step-by-step explanation:

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.

Answer 2

Step-by-step explanation:

A×B =

$ab = {x}^{2} + xy + {y}^{2} \times (x - y)$

$= {x}^{3} - {x}^{2} y + {x}^{2} y - x {y}^{2} + x {y}^{2} - {y}^{3}$

${x}^{3} - {y}^{3}$

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