Solve frac{a^{2}-ab+ab^{2}}{a}+b^2-bc+bc^2/b+c^2-ca+ca^2/c

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

Evaluate

$$a^{2}+b^{2}+c^{2}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}-ab+ab^{2}}{a}.

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

Cancel out a in both numerator and denominator.

$$a+b^{2}-b+\frac{b^{2}-bc+bc^{2}}{b}+\frac{c^{2}-ca+ca^{2}}{c}$$

Factor the expressions that are not already factored in \frac{b^{2}-bc+bc^{2}}{b}.

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

Cancel out b in both numerator and denominator.

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

Combine -b and b to get 0.

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

Factor the expressions that are not already factored in \frac{c^{2}-ca+ca^{2}}{c}.

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

Cancel out c in both numerator and denominator.

$$a+b^{2}+c^{2}-c+a^{2}-a+c$$

Combine a and -a to get 0.

$$b^{2}+c^{2}-c+a^{2}+c$$

Combine -c and c to get 0.

$$b^{2}+c^{2}+a^{2}$$

Expand

$$a^{2}+b^{2}+c^{2}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{2}-ab+ab^{2}}{a}.

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

Cancel out a in both numerator and denominator.

$$a+b^{2}-b+\frac{b^{2}-bc+bc^{2}}{b}+\frac{c^{2}-ca+ca^{2}}{c}$$

Factor the expressions that are not already factored in \frac{b^{2}-bc+bc^{2}}{b}.

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

Cancel out b in both numerator and denominator.

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

Combine -b and b to get 0.

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

Factor the expressions that are not already factored in \frac{c^{2}-ca+ca^{2}}{c}.

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

Cancel out c in both numerator and denominator.

$$a+b^{2}+c^{2}-c+a^{2}-a+c$$

Combine a and -a to get 0.

$$b^{2}+c^{2}-c+a^{2}+c$$

Combine -c and c to get 0.

$$b^{2}+c^{2}+a^{2}$$