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已知X的4次方+Y的4次方=50,X的2次方Y-XY的2次方=14,求X的4次方-Y的4次方+3XY的2次方-XY的2次方-5X的2次方Y+3的2次方+2Y的4次方
题目详情
已知X的4次方+Y的4次方=50,X的2次方Y-XY的2次方=14,求X的4次方-Y的4次方+3XY的2次方-XY的2次方-5X的2次方Y+3的2次方+2Y的4次方
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答案和解析
x^4+y^4=50 => (x^2-y^2)^2+2x^2y^2=50
=> (x+y)^2(x-y)^2+2x^2y^2=50
x^2y-xy^2=14 => xy(x-y)=14
x^4-y^4+3xy^2-xy^2-5x^2y+3x^2y+2y^4
=(x^4-y^4+2y^4)+(3xy^2-xy^2-5x^2y+3x^y)
=(x^4+y^4)+(2xy^2-2x^2y)
=(x^4+y^4)-2(x^2y-xy^2)
=50-14×2
=50-28
=22
=> (x+y)^2(x-y)^2+2x^2y^2=50
x^2y-xy^2=14 => xy(x-y)=14
x^4-y^4+3xy^2-xy^2-5x^2y+3x^2y+2y^4
=(x^4-y^4+2y^4)+(3xy^2-xy^2-5x^2y+3x^y)
=(x^4+y^4)+(2xy^2-2x^2y)
=(x^4+y^4)-2(x^2y-xy^2)
=50-14×2
=50-28
=22
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