因式分解(2x-y)四次方+(x+y)四次方+(2y-x)四次方

题目

因式分解(2x-y)四次方+(x+y)四次方+(2y-x)四次方

答案

令a=2x-y ,b=2y-x ,a+b=(2x-y)+(2y-x)=x+y
(2x-y)四次方+(x+y)四次方+(2y-x)四次方
=a四次方+(a+b)四次方+b四次方
=(a+b)四次方+( a四次方+2a²b²+b四次方)- 2a²b²
=(a+b)四次方+(a²+b²)² - 2a²b²
=(a+b)四次方+[ (a+b)²-2ab ]² - 2a²b²
=(a+b)四次方+[ (a+b)四次方-4ab(a+b)²+4a²b² ] - 2a²b²
=2(a+b)四次方-4ab(a+b)²+2a²b²
=2[ (a+b)四次方-2ab(a+b)²+(ab)² ]
=2[ (a+b)²-(ab) ]²
=2[ (x+y)²-(2x-y)(2y-x) ]²
=2(x²+2xy+y²-5xy+2x²+2y²)²
=2(3x²+3y²-3xy)²
=18(x²+y²-xy)²