Question

$x^{6}y^{3}+y^{9}=$

Explanation:

Step1: Factor out common term

Identify and extract the greatest common factor $y^3$ from both terms.
$x^6y^3 + y^9 = y^3(x^6 + y^6)$

Step2: Rewrite terms as perfect cubes

Express $x^6$ as $(x^2)^3$ and $y^6$ as $(y^2)^3$ to use sum of cubes identity.
$x^6 + y^6 = (x^2)^3 + (y^2)^3$

Step3: Apply sum of cubes formula

Use the identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ where $a=x^2$, $b=y^2$.
$(x^2)^3 + (y^2)^3 = (x^2 + y^2)(x^4 - x^2y^2 + y^4)$

Step4: Combine all factors

Substitute back into the factored expression from Step1.
$y^3(x^2 + y^2)(x^4 - x^2y^2 + y^4)$

Answer:

$y^3(x^2 + y^2)(x^4 - x^2y^2 + y^4)$