Toán Lớp 8: Chứng minh đẳng thức:
$a)$$\frac{y}{(x-y)(y-z)}+$ $\frac{z}{(y-z)(z-x)}+$ $\frac{x}{(z-x)(x-y)}=0$
$b)$ $\frac{x^{2}}{(x-y)(x-z)}+$ $\frac{y^{2}}{(y-z)(y-x)}+$ $\frac{z^{2}}{(z-x)(z-y)}=1$
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Toán Lớp 8: Chứng minh đẳng thức:
$a)$$\frac{y}{(x-y)(y-z)}+$ $\frac{z}{(y-z)(z-x)}+$ $\frac{x}{(z-x)(x-y)}=0$
$b)$ $\frac{x^{2}}{(x-y)(x-z)}+$ $\frac{y^{2}}{(y-z)(y-x)}+$ $\frac{z^{2}}{(z-x)(z-y)}=1$
Comments ( 1 )
$a)$ $\dfrac{y}{\left(x-y\right)\left(y-z\right)}+\dfrac{z}{\left(y-z\right)\left(z-x\right)}+\dfrac{x}{\left(z-x\right)\left(x-y\right)}=0$
Ta có:
VT = $\dfrac{y}{\left(x-y\right)\left(y-z\right)}+\dfrac{z}{\left(y-z\right)\left(z-x\right)}+\dfrac{x}{\left(z-x\right)\left(x-y\right)}$
$=\dfrac{y\left(z-x\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}+\dfrac{z\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}+\dfrac{x\left(y-z\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}$
$=\dfrac{y\left(z-x\right)+z\left(x-y\right)+x\left(y-z\right)}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}$
$=\dfrac{0}{\left(x-y\right)\left(y-z\right)\left(z-x\right)}$
$=0$ = VP (đpcm)
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$b)$ $\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-z\right)\left(y-x\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}=1$
Ta có:
VT= $\dfrac{x^2}{\left(x-y\right)\left(x-z\right)}+\dfrac{y^2}{\left(y-z\right)\left(y-x\right)}+\dfrac{z^2}{\left(z-x\right)\left(z-y\right)}$
$=\dfrac{x^2\left(-y+z\right)}{-\left(x-y\right)\left(x-z\right)\left(y-z\right)}+\dfrac{y^2\left(-z+x\right)}{-\left(x-y\right)\left(x-z\right)\left(y-z\right)}+\dfrac{z^2\left(-x+y\right)}{-\left(x-y\right)\left(x-z\right)\left(y-z\right)}$
$=\dfrac{x^2\left(-y+z\right)+y^2\left(-z+x\right)+z^2\left(-x+y\right)}{-\left(x-y\right)\left(x-z\right)\left(y-z\right)}$
$=-\dfrac{-x^2y+x^2z-y^2z+xy^2-xz^2+yz^2}{x^2z-x^2y+xy^2-xz^2+yz^2-y^2z}$
$=-\left(-1\right)$
$=1$ = VP (đpcm)