Find the greatest common factor of x^3y-xy^3 and x(x-y)^2
You asked:
Find the polynomial greatest common divisor of the following expressions: \(x^{3} y - y^{3} x\), \(\left(x \left(x - y\right)\right)^{2}\)
MathBot Answer:
The greatest common divisor is \(x^{2} - x y\).
Begin by factoring all expressions completely. $$\begin{aligned}x^{3} y - y^{3} x &= x y \left(x - y\right) \left(x + y\right)\\\left(x \left(x - y\right)\right)^{2} &= x^{2} \left(x - y\right)^{2}\\\end{aligned}$$
Find the greatest common divisor by multiplying common factors together, including only the lowest power of those factors which are common to all expressions. \[\operatorname{gcd}(x^{3} y - y^{3} x, \left(x \left(x - y\right)\right)^{2}) = (x - y)^{1}\cdot (x)^{1} = x^{2} - x y\]