Question

Cho x² - y = a , y² - z = b , z² - x = c

Với a,b,c là các hằng số . CMR: các biểu thức sau k phụ thuộc vào biến x,y,z

P = x³ ( z - y² ) + y³( x - z² ) + z³( y - x² ) + xyz ( xyz - 1 )

Answer 1

P = ...

\(\Leftrightarrow P=x^3z-x^3y^2+y^3x-y^3z^2+z^3y-z^3x^2+x^2y^2z^2-xyz\)\(\Leftrightarrow P=\left(x^3z-x^2z^3\right)-\left(x^3y^2-x^2y^2z^2\right)+\left(xy^3-y^3z\right)+\left(yz^3-xyz\right)\)\(\Leftrightarrow P=x^2z\left(x-z^2\right)-x^2y^2\left(x-z^2\right)+y^3\left(x-z^2\right)-yz\left(x-z^2\right)\)\(\Leftrightarrow P=\left(x-z^2\right)\left(x^2z-x^2y^2+y^3-yz\right)\)

\(\Leftrightarrow P=\left(x-z^2\right)\left[\left(x^2z-x^2y^2\right)+\left(y^3-yz\right)\right]\)

\(\Leftrightarrow P=\left(x-z^2\right)\left[-x^2\left(y^2-z\right)+y\left(y^2-z\right)\right]\)

\(\Leftrightarrow P=\left(x-z\right)^2\left(y^2-z\right)\left(y-x^2\right)\)

\(\Leftrightarrow P=abc\left(đpcm\right)\)