题目内容
若x+y=-1,则x4+5x3y+x2y+8x2y2+xy2+5xy3+y4的值等于
- A.0
- B.-1
- C.1
- D.3
C
分析:将x4+5x3y+x2y+8x2y2+xy2+5xy3+y4提取公因式,转化为x3(x+y)+4x2y(x+y)+xy(x+y)+4xy2(x+y)+y3(x+y),将x+y=-1代入转化为-x3-4x2y-xy-4xy2-y3,再通过提取公因式,立方和公式、提取公因式,转化为(x+y)(x2-xy+y2)-4xy+xy,再将x+y=-1代入转化为(x2-xy+y2)+3xy,再运用完全平方式,转化为(x+y)2-3xy+3xy,将x+y=-1代入问题得解.
解答:原式=x4+x3y+4x3y+x2y+4x2y2+4x2y2+xy2+4xy3+xy3+y4,
=x3(x+y)+4x2y(x+y)+xy(x+y)+4xy2(x+y)+y3(x+y),
=-x3-4x2y-xy-4xy2-y3,
=-[(x3+y3)+4xy(x+y)+xy],
=-[(x+y)(x2-xy+y2)-4xy+xy],
=-[-(x2-xy+y2)-3xy],
=(x2-xy+y2)+3xy,
=(x+y)2-3xy+3xy,
=1.
故选C.
点评:本题考查提取公因式法因式分解、完全平方式、立方和公式.解决本题的关键是提取公因式,转化为(x+y)乘以xy各次的形式,逐步达到化简得目的.
分析:将x4+5x3y+x2y+8x2y2+xy2+5xy3+y4提取公因式,转化为x3(x+y)+4x2y(x+y)+xy(x+y)+4xy2(x+y)+y3(x+y),将x+y=-1代入转化为-x3-4x2y-xy-4xy2-y3,再通过提取公因式,立方和公式、提取公因式,转化为(x+y)(x2-xy+y2)-4xy+xy,再将x+y=-1代入转化为(x2-xy+y2)+3xy,再运用完全平方式,转化为(x+y)2-3xy+3xy,将x+y=-1代入问题得解.
解答:原式=x4+x3y+4x3y+x2y+4x2y2+4x2y2+xy2+4xy3+xy3+y4,
=x3(x+y)+4x2y(x+y)+xy(x+y)+4xy2(x+y)+y3(x+y),
=-x3-4x2y-xy-4xy2-y3,
=-[(x3+y3)+4xy(x+y)+xy],
=-[(x+y)(x2-xy+y2)-4xy+xy],
=-[-(x2-xy+y2)-3xy],
=(x2-xy+y2)+3xy,
=(x+y)2-3xy+3xy,
=1.
故选C.
点评:本题考查提取公因式法因式分解、完全平方式、立方和公式.解决本题的关键是提取公因式,转化为(x+y)乘以xy各次的形式,逐步达到化简得目的.
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