Solve (b-c)^2/(a-b)(a-c)+(c-a)^2/(b-c)(b-a)+(a-b)^2/(c-a)(c-b)

$$\frac { ( b - c ) ^ { 2 } } { ( a - b ) ( a - c ) } + \frac { ( c - a ) ^ { 2 } } { ( b - c ) ( b - a ) } + \frac { ( a - b ) ^ { 2 } } { ( c - a ) ( c - b ) }$$

Evaluate

$$-3$$

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-c\right)\left(-a+b\right)\left(b-c\right). Multiply \frac{\left(b-c\right)^{2}}{\left(a-b\right)\left(a-c\right)} times \frac{-\left(b-c\right)}{-\left(b-c\right)}. Multiply \frac{\left(c-a\right)^{2}}{\left(b-c\right)\left(b-a\right)} times \frac{a-c}{a-c}.

$$\frac{\left(b-c\right)^{2}\left(-1\right)\left(b-c\right)}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}+\frac{\left(c-a\right)^{2}\left(a-c\right)}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}+\frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)}$$

Since \frac{\left(b-c\right)^{2}\left(-1\right)\left(b-c\right)}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)} and \frac{\left(c-a\right)^{2}\left(a-c\right)}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(b-c\right)^{2}\left(-1\right)\left(b-c\right)+\left(c-a\right)^{2}\left(a-c\right)}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}+\frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)}$$

Do the multiplications in \left(b-c\right)^{2}\left(-1\right)\left(b-c\right)+\left(c-a\right)^{2}\left(a-c\right).

$$\frac{-b^{3}+b^{2}c+2b^{2}c-2bc^{2}-c^{2}b+c^{3}+c^{2}a-c^{3}-2ca^{2}+2c^{2}a+a^{3}-a^{2}c}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}+\frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)}$$

Combine like terms in -b^{3}+b^{2}c+2b^{2}c-2bc^{2}-c^{2}b+c^{3}+c^{2}a-c^{3}-2ca^{2}+2c^{2}a+a^{3}-a^{2}c.

$$\frac{-b^{3}-3bc^{2}+3b^{2}c-3ca^{2}+3c^{2}a+a^{3}}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}+\frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)}$$

Factor the expressions that are not already factored in \frac{-b^{3}-3bc^{2}+3b^{2}c-3ca^{2}+3c^{2}a+a^{3}}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}.

$$\frac{\left(-a+b\right)\left(-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}\right)}{\left(a-c\right)\left(-a+b\right)\left(b-c\right)}+\frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)}$$

Cancel out -a+b in both numerator and denominator.

$$\frac{-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}}{\left(a-c\right)\left(b-c\right)}+\frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)}$$

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(b-c\right)\left(-a+c\right). Multiply \frac{-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}}{\left(a-c\right)\left(b-c\right)} times \frac{-1}{-1}. Multiply \frac{\left(a-b\right)^{2}}{\left(c-a\right)\left(c-b\right)} times \frac{-1}{-1}.

$$\frac{-\left(-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}\right)}{\left(b-c\right)\left(-a+c\right)}+\frac{-\left(a-b\right)^{2}}{\left(b-c\right)\left(-a+c\right)}$$

Since \frac{-\left(-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}\right)}{\left(b-c\right)\left(-a+c\right)} and \frac{-\left(a-b\right)^{2}}{\left(b-c\right)\left(-a+c\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}\right)-\left(a-b\right)^{2}}{\left(b-c\right)\left(-a+c\right)}$$

Do the multiplications in -\left(-a^{2}-ab+3ac-b^{2}+3bc-3c^{2}\right)-\left(a-b\right)^{2}.

$$\frac{a^{2}+ab-3ac+b^{2}-3bc+3c^{2}-a^{2}+2ab-b^{2}}{\left(b-c\right)\left(-a+c\right)}$$

Combine like terms in a^{2}+ab-3ac+b^{2}-3bc+3c^{2}-a^{2}+2ab-b^{2}.

$$\frac{3ab-3ac-3bc+3c^{2}}{\left(b-c\right)\left(-a+c\right)}$$

Factor the expressions that are not already factored in \frac{3ab-3ac-3bc+3c^{2}}{\left(b-c\right)\left(-a+c\right)}.

$$\frac{3\left(a-c\right)\left(b-c\right)}{\left(b-c\right)\left(-a+c\right)}$$

Extract the negative sign in -c+a.

$$\frac{-3\left(b-c\right)\left(-a+c\right)}{\left(b-c\right)\left(-a+c\right)}$$

Cancel out \left(b-c\right)\left(-a+c\right) in both numerator and denominator.

$$-3$$

Factor

$$-3$$