x^3y^2+512y^2
x^3y^2 + 512y^2 = y^2(x^3 + 512)
= y^2 (x^3 + 8^3)
= y^2 (x+8) (x^2 - 8x + 64)
To simplify the expression x^3y^2 + 512y^2, we can start by factoring out a common factor of y^2.
Step 1: Factor out y^2:
y^2(x^3 + 512)
Next, we notice that x^3 + 512 is the sum of two cubes. The sum of two cubes can be factored using the formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Step 2: Apply the sum of cubes formula:
y^2(x + 8)(x^2 - 8x + 64)
Therefore, the fully simplified expression is y^2(x + 8)(x^2 - 8x + 64).