Problem 140 - Algebra and Trigonometry

Question

Factor completely. $$x^{4}-y^{4}-2 x^{3} y+2 x y^{3}$$

Step-by-Step Solution

Answer

The factorization of the original polynomial is \((x^{2} - y^{2} - xy)(x^{2} - y^{2} + xy)\).

Step 1: Rearrange the Terms

Rearrange the terms in such a way that we get two difference of squares: \(x^{4} -2 x^{3} y + 2x y^{3} - y^{4}\). This is simply the same polynomial as above but rearranged.

Step 2: Factor Using Difference of Squares

Take first two and last two terms separately and factor them using the formula for difference of squares. On applying, we get \((x^{2} - y^{2})^{2} - (x y)^{2}\).

Step 3: Apply Difference of Squares Again

Recognize this as another difference of squares, and apply the formula again: \((x^{2} - y^{2} - xy)(x^{2} - y^{2} + xy)\). The polynomial is now factored completely.