Question

Factorie completely x^5y^2-xy^6

Answer 1

To factor the expression \( x^5y^2 - xy^6 \) completely, we first look for common factors in both terms.

1. Identify the common factors:
- The first term is \( x^5y^2 \).
- The second term is \( -xy^6 \).
- The common factor in both terms is \( xy^2 \).

2. Factor out the common factor \( xy^2 \):

\[
x^5y^2 - xy^6 = xy^2(x^4 - y^4)
\]

3. Now, further factor \( x^4 - y^4 \):
- This is a difference of squares, which can be factored as follows:

\[
x^4 - y^4 = (x^2 - y^2)(x^2 + y^2)
\]

4. Next, we can also factor \( x^2 - y^2 \) further since it is also a difference of squares:

\[
x^2 - y^2 = (x - y)(x + y)
\]

5. Putting everything together, we have:

\[
x^5y^2 - xy^6 = xy^2(x^4 - y^4) = xy^2((x - y)(x + y)(x^2 + y^2))
\]

Therefore, the completely factored form of the expression \( x^5y^2 - xy^6 \) is:

\[
xy^2(x - y)(x + y)(x^2 + y^2)
\]