Question

If ${(x}^4+x^{2}y+y^2)$ is one of the factors of an expression which is the difference of two cubes, then the other factor is ___.

• A. $x^2-y$
• B. $x^2-y^2$
• C. $x-y^2$
• D. $x-y$

Answer 1

The correct option is A $x^2-y$

Let two numbers be a and b.
The difference of their cubes is $a^3–b^3$ and hence this is the expression we need.

This expression can be factorised as $a^3–b^3=(a-b)(a^2+ab+b^2)$.

Now, consider the given factor, $x^4+x^{2}y+y^2={\left(x^2\right)}^2+x^{2}y+{\left(y\right)}^2$

This is in the form of $a^2+ab+b^2$ where $a=x^2,b=y$, which is similar to the factor given.

Therefore, the other factor is of the form $a–b$.

Hence, other factor is $x^2-y.$