Solve (a-b)(a+b)+(b-c)(b+c)+(c-a)(c+a)

$$( a - b ) ( a + b ) + ( b - c ) ( b + c ) + ( c - a ) ( c + a )$$

Evaluate

$$0$$

Solution Steps

Consider \left(a-b\right)\left(a+b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Consider \left(b-c\right)\left(b+c\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Combine -b^{2} and b^{2} to get 0.

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

Consider \left(c-a\right)\left(c+a\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Combine -c^{2} and c^{2} to get 0.

$$a^{2}-a^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$0$$

Factor

$$0$$