Let $x+y=a$ and $xy=b$.  The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$.  What is this polynomial?
1. Given the equations \(x + y = a\) and \(xy = b\), we need to express \(x^6 + y^6\) in terms of \(a\) and \(b\).

2. First, recall the identity for the sum of cubes:
   \[
   x^3 + y^3 = (x+y)(x^2 - xy + y^2)
   \]
   Using \(x + y = a\) and \(xy = b\), we can rewrite \(x^2 + y^2\) as follows:
   \[
   x^2 + y^2 = (x+y)^2 - 2xy = a^2 - 2b
   \]
   Therefore,
   \[
   x^3 + y^3 = a(x^2 - xy + y^2) = a(a^2 - 3b)
   \]
   Hence,
   \[
   x^3 + y^3 = a^3 - 3ab
   \]

3. Next, we need to find \(x^6 + y^6\). We use the identity for the square of the sum of cubes:
   \[
   (x^3 + y^3)^2 = x^6 + y^6 + 2x^3y^3
   \]
   Substituting \(x^3 + y^3 = a^3 - 3ab\) and \(x^3y^3 = (xy)^3 = b^3\), we get:
   \[
   (a^3 - 3ab)^2 = x^6 + y^6 + 2b^3
   \]

4. Solving for \(x^6 + y^6\), we have:
   \[
   x^6 + y^6 = (a^3 - 3ab)^2 - 2b^3
   \]

5. Expanding \((a^3 - 3ab)^2\):
   \[
   (a^3 - 3ab)^2 = a^6 - 2 \cdot 3a^3 \cdot 3ab + (3ab)^2 = a^6 - 6a^4b + 9a^2b^2
   \]

6. Therefore:
   \[
   x^6 + y^6 = a^6 - 6a^4b + 9a^2b^2 - 2b^3
   \]

The final answer is \(\boxed{a^6 - 6a^4b + 9a^2b^2 - 2b^3}\).