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Recognize a perfect square in three variables

Three squares and three doubled products form one square

Use $A^{2} + B^{2} + C^{2} + 2 A B + 2 B C + 2 C A = (A + B + C)^{2}$ . The key is to identify the same three quantities in every term.

Problem

Factor

(x + 1)^{2} + (y - 1)^{2} + (z - 2)^{2} + 2 (x y - x + y - 1) + 2 (y z - 2 y - z + 2) + 2 (z x - 2 x + z - 2) .

Answer

(x + y + z - 2)^{2}

. In particular,

x + y + z - 2

is a factor.

The three bracketed expressions are products of the same quantities that appear in the squares.

Step-by-step solution

1. Name the three quantities

Let $A = x + 1$ , $B = y - 1$ , and $C = z - 2$ .

2. Identify the mixed products

The three bracketed expressions are

x y - x + y - 1 = (x + 1) (y - 1) = A B,

y z - 2 y - z + 2 = (y - 1) (z - 2) = B C,

z x - 2 x + z - 2 = (z - 2) (x + 1) = C A .

3. Apply the identity

The original is $A^{2} + B^{2} + C^{2} + 2 A B + 2 B C + 2 C A$ , hence $(A + B + C)^{2}$ . The sum is $(x + 1) + (y - 1) + (z - 2) = x + y + z - 2$ , so the answer is $(x + y + z - 2)^{2}$ .

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