Question

Cho x+y=2;x^2+y^2=b;x^3+y^3=c

Chứng minh a^3-3ab+2c=0

Answer 1

Sửa đề: Cho \(x+y=a;x^2+y^2=b;x^3+y^3=c\)

Chứng minh: \(a^3-2ab+2c=0\)

Giải:

Ta có:

\(a^3-3ab+2c=\left(x+y\right)^3-3\left(x+y\right)\left(x^2+y^2\right)+2\left(x^3+y^3\right)\)

\(=x^3+y^3+3xy\left(x+y\right)-3\left(x+y\right)\left(x^2+y^2\right)+2\left(x^3+y^3\right)\)

\(=3\left(x^3+y^3\right)+3\left(x+y\right)\left(xy-x^2-y^2\right)=3\left(x+y\right)\left(x^2-xy+y^2\right)+3\left(x+y\right)\left(xy-x^2-y^2\right)\)

\(=3\left(x+y\right)\left(x^2-xy+y^2+xy-x^2-y^2\right)=3\left(x+y\right).0\)

\(=0\) (đpcm)