Question

Cho x+y=a+b; x^2+y^2=a^2+b^2. Chứng minh rằng x^3+y^3=a^3+b^3

Answer 1

ta có : \(x^2+y^2=a^2+b^2\Leftrightarrow x^2+2xy+y^2-2xy=a^2+2ab+b^2-2ab\)

\(\Leftrightarrow\left(x+y\right)^2-2xy=\left(a+b\right)^2-2ab\) (vì : \(x+y=a+b\))

\(\Rightarrow-2xy=-2ab\Leftrightarrow xy=ab\)

ta có : \(x+y=a+b\Leftrightarrow\left(x+y\right)^3=\left(a+b\right)^3\)

\(\Leftrightarrow x^3+3x^2y+3xy^2+y^3=a^3+3a^2b+3ab^2+b^3\)

\(\Leftrightarrow x^3+y^3+3xy\left(x+y\right)=a^3+b^3+3ab\left(a+b\right)\)

(vì : \(x+y=a+bvàxy=ab\))

\(\Rightarrow x^3+y^3=a^3+b^3\) (đpcm)