Key idea. The expression is almost the expansion of a cube.

1. Expand:

   \[(x+y)^3=x^3+3x^2y+3xy^2+y^3.\]

2. Factor the two middle terms: \(3x^2y+3xy^2=3xy(x+y)\). Since \(x+y=1\), these terms together equal \(3xy\).

3. Thus \(x^3+y^3+3xy=(x+y)^3=1^3=1\).

Remember. Before substituting individual numbers, compare an expression with a familiar identity.