simplify x/(y-x)(z-x) + y/(z-y)(x-y) + z/(x-z)(y-z)
*Hint: the common denominator consists of 3 factors only
Please show all important steps
You need to change the sign on some of the factors, so the factor is the same in each fraction,
for instance
$\displaystyle (y-x)=-(x-y),\ (z-x)=-(x-z),\ (z-y)=-(y-z)$
$\displaystyle \frac{x}{(y-x)(z-x)}+\frac{y}{(z-y)(x-y)}+\frac{z}{(x-z)(y-z)}$
$\displaystyle =\frac{x}{(x-y)(x-z)}+\frac{-y}{(x-y)(y-z)}+\frac{z}{(x-z)(y-z)}$
Next you check which fractions have which factor missing, then multiply the fraction by
$\displaystyle \frac{missing\ factor}{missing\ factor}$
for example, the first fraction must be multiplied by $\displaystyle \frac{(y-z)}{(y-z)}$