已知：X+Y+Z=a,X^2+Y^2+Z^2=b^2,X^3+Y^3+Z^3-3XYZ=c^3,求证3ab^2=a^3+2c^3_数学解答

已知：X+Y+Z=a,X^2+Y^2+Z^2=b^2,X^3+Y^3+Z^3-3XYZ=c^3,求证3ab^2=a^3+2c^3

人气：292 ℃　时间：2020-04-03 22:23:59

解答

右边=a^3+2c^3=（X+Y+Z）^3+2(X^3+Y^3+Z^3-3XYZ)
=3(x^3+y^3+Z^3+x^2y+x^2z+y^2x+y^2z+z^2x+z^2y),
左边=3ab^2=3(X+Y+Z)(X^2+Y^2+Z^2)
=3(x^3+y^3+Z^3+x^2y+x^2z+y^2x+y^2z+z^2x+z^2y),
所以原等式成立