Fill in the blanks:
(x2−y)2+(y2−x)2=(?)×(x2+1)+2xy×(?)+y2×(?)
To solve this problem, let's consider the expansion of the expressions:
First, expand each square:
- (x2−y)2=(x2)2−2(x2)y+y2
- (y2−x)2=(y2)2−2(y2)x+x2
Expanding these expressions, we get:
(x2−y)2=x4−2x2y+y2
(y2−x)2=y4−2y2x+x2
Adding the two expanded forms gives:
x4−2x2y+y2+y4−2y2x+x2=x4+y4+x2+y2−2x2y−2y2x
Now rearrange the terms in a form similar to the given expression:
The goal is to arrange the polynomial as:
a(x2+1)+2xbcxy+c(y2+1).
Since now we want:
(x2−y)2+(y2−x)2=a(x2+1)+2xy(−(x+y))+c(y2+1).
Create matching forms:
- The term for a(x2+1) will have a=x2.
- The term for b in 2xy(−(x+y)) will have b=−(x+y).
- The term for c(y2+1) will have c=y2+1.
Therefore, the correct values to fill in are (x2,−(x+y),y2+1).
Thus, our complete expression is:
(x2−y)2+(y2−x)2=x2(x2+1)+2xy(−(x+y))+y2(y2+1)
Hence, the answer is:
x2,−(x+y),y2+1
x2,−(x+y),y2+1