Solve 1/(a-b)(a-c)+1/(b-c)(b-a)+1/(c-a)(c-b)=

$$\frac { 1 } { ( a - b ) ( a - c ) } + \frac { 1 } { ( b - c ) ( b - a ) } + \frac { 1 } { ( c - a ) ( c - b ) } =$$

Evaluate

$$0$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-b\right)\left(a-c\right) and \left(b-c\right)\left(b-a\right) is \left(a-b\right)\left(a-c\right)\left(-b+c\right). Multiply \frac{1}{\left(a-b\right)\left(a-c\right)} times \frac{-b+c}{-b+c}. Multiply \frac{1}{\left(b-c\right)\left(b-a\right)} times \frac{a-c}{a-c}.

$$\frac{-b+c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}+\frac{a-c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}$$

Since \frac{-b+c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)} and \frac{a-c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)} have the same denominator, add them by adding their numerators.

$$\frac{-b+c+a-c}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}$$

Combine like terms in -b+c+a-c.

$$\frac{-b+a}{\left(a-b\right)\left(a-c\right)\left(-b+c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}$$

Cancel out a-b in both numerator and denominator.

$$\frac{1}{\left(a-c\right)\left(-b+c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-c\right)\left(-b+c\right) and \left(c-a\right)\left(c-b\right) is \left(-a+c\right)\left(-b+c\right). Multiply \frac{1}{\left(a-c\right)\left(-b+c\right)} times \frac{-1}{-1}.

$$\frac{-1}{\left(-a+c\right)\left(-b+c\right)}+\frac{1}{\left(-a+c\right)\left(-b+c\right)}$$

Since \frac{-1}{\left(-a+c\right)\left(-b+c\right)} and \frac{1}{\left(-a+c\right)\left(-b+c\right)} have the same denominator, add them by adding their numerators. Add -1 and 1 to get 0.

$$\frac{0}{\left(-a+c\right)\left(-b+c\right)}$$

Zero divided by any non-zero term gives zero.

$$0$$

Factor

$$0$$