Question

已知x+y=1，求证x立方+3xy+y立方=1

Answer 1

解析，
x³+y³+3xy
=(x+y)(x²+y²-xy)+3xy
=(x+y)[(x+y)²-3xy]+3xy
x+y=1
因此，
x³+y³+3xy=1-3xy+3xy=1

Answer 2

x立方+3xy+y立方=(X+Y)(X平方-XY+Y平方）+3XY
因为x+y=1
所以原式=X平方-XY+Y平方+3XY
= X平方+2XY+Y平方
=(X+Y)平方
=1
x立方+3xy+y立方=1

Answer 3

x立方+3xy+y立方
=(x³+y³)+3xy
=(x+y)(x²-xy+y²)+3xy
因为：x+y=1
原式=x²-xy+y²+3xy
=x²+2xy+y²
=(x+y)²
=1

Answer 4

证明 因为 x+y=1
所以 (x+y)^3=x^3+3x^2y+3xy^2+y^3=1

(x^3+3xy+y^3)-(x^3+3x^2y+3xy^2+y^3)
=3xy-3x^2y-3xy^2
=3xy(1-x-y)
=3xy*0
=0
所以x^3+3x^2y+3xy^2+y^3=x立方+3xy+y立方=1

Answer 5

x³＋3xy＋y³
=(x³＋y³)＋3xy
=(x＋y)(x²－xy＋y²)＋3xy
=x²－2xy＋y²＋3xy
=x²＋2xy＋y²
=(x＋y)²
=1