Complete the Perfect Square Expression: (x²-y)² + (y²-x)² Expansion Puzzle

Fill in the blanks:

$(x^2-y)^2+(y^2-x)^2=(?)\times(x^2+1)+2xy\times(?)+y^2\times(?)$

To solve this problem, let's consider the expansion of the expressions:

First, expand each square:

• $(x^2 - y)^2 = (x^2)^2 - 2(x^2)y + y^2$
• $(y^2 - x)^2 = (y^2)^2 - 2(y^2)x + x^2$

Expanding these expressions, we get:

$(x^2 - y)^2 = x^4 - 2x^2y + y^2$
$(y^2 - x)^2 = y^4 - 2y^2x + x^2$

Adding the two expanded forms gives:

$x^4 - 2x^2y + y^2 + y^4 - 2y^2x + x^2 = x^4 + y^4 + x^2 + y^2 - 2x^2y - 2y^2x$

Now rearrange the terms in a form similar to the given expression:

The goal is to arrange the polynomial as: $a(x^2 + 1) + 2xbcxy + c(y^2 + 1)$.

Since now we want: $(x^2 - y)^2 + (y^2 - x)^2 = a(x^2 + 1) + 2xy(- (x+y)) + c(y^2 + 1)$.

Create matching forms:

• The term for $a(x^2 + 1)$ will have $a = x^2$.
• The term for $b$ in $2xy(- (x+y))$ will have $b = -(x+y)$.
• The term for $c(y^2 + 1)$ will have $c = y^2 + 1$.

Therefore, the correct values to fill in are $(x^2,-(x+y),y^2+1)$.

Thus, our complete expression is:

$(x^2 - y)^2 + (y^2 - x)^2 = x^2(x^2+1) + 2xy(-(x+y)) + y^2(y^2+1)$

Hence, the answer is:

$x^2,-(x+y),y^2+1$