Question

The value of ${(x-y)}^3+{(x+y)}^3+3{(x-y)}^2(x+y)+3{(x+y)}^2(x-y)$ is

• A. $27x^3$
• B. $8(x^3-y^3)$
• C. $8x^3{y}^3$
• D. $8x^3$

Answer 1

Answer: (D)

$8x^3$

Since $(a+b{)}^3=a^3+b^3+3a^{2}b+3a{b}^2$,

In the expression $(x-y{)}^3+(x+y{)}^3+3(x-y{)}^2(x+y)+3(x+y{)}^2(x-y)$,

$a=x-y$ and $b=x+y$

Therefore, using the identity $(a+b{)}^3=a^3+b^3+3a^{2}b+3a{b}^2$, the expression $(x-y{)}^3+(x+y{)}^3+3(x-y{)}^2(x+y)+3(x+y{)}^2(x-y)$ can be written as:

$(x-y{)}^3+(x+y{)}^3+3(x-y{)}^2(x+y)+3(x+y{)}^2(x-y)=\left[\right(x-y)+(x+y){]}^3=(2x{)}^3=8x^3$

Hence, $(x-y{)}^3+(x+y{)}^3+3(x-y{)}^2(x+y)+3(x+y{)}^2(x-y)=8x^3$.