Problem
ALG-B1-M01-P007 Expression Through x+y
#7
★★☆☆☆ Level 2 of 5
If \(x+y=1\), find \(x^3+y^3+3xy\).
Use \(x^3+y^3=(x+y)^3-3xy(x+y)\).
By the sum of cubes identity, \[ x^3+y^3=(x+y)^3-3xy(x+y). \] Since \(x+y=1\), we get \(x^3+y^3=1-3xy\).
Thus \(x^3+y^3+3xy=1\).
Shows that sometimes an expression does not depend on \(xy\).