分解因式:(1)(2x2﹣3x+1)2﹣22x2+33x﹣1;(2)x4+7x3+14x2+7x+1;(3)(x+y)3+2xy(1﹣x﹣y)﹣1;(4)(x+
题型:解答题难度:简单来源:不详
分解因式: (1)(2x2﹣3x+1)2﹣22x2+33x﹣1; (2)x4+7x3+14x2+7x+1; (3)(x+y)3+2xy(1﹣x﹣y)﹣1; (4)(x+3)(x2﹣1)(x+5)﹣20. |
答案
(1)x(2x﹣3)(2x+3)(x﹣3) (2)(x2+4x+1)(x2+3x+1) (3)(x+y﹣1)(x2+y2+x+y+1) (4)(x2+4x+5)(x2+4x﹣7) |
解析
试题分析:(1)把2x2﹣3x+1看成整体,构造和它有关的式子,然后进一步分解; (2)由x的最高指数,联想到[(x+1)2]2,努力构造这个形式解答; (3)第一、三项符合立方差公式,再提取公因式即可; (4)把原式化为(x+3)(x+1)(x﹣1)(x+5)﹣20=(x2+4x+3)(x2+4x﹣5)﹣20,把x2+4x看成整体解答. 解:(1)(2x2﹣3x+1)2﹣22x2+33x﹣1, =(2x2﹣3x+1)2﹣11(2x2﹣3x+1)+10, =(2x2﹣3x+1﹣1)(2x2﹣3x+1﹣10), =(2x2﹣3x)(2x2﹣3x﹣9), =x(2x﹣3)(2x+3)(x﹣3); (2)x4+7x3+14x2+7x+1, =x4+4x3+6x2+4x+1+3x3+6x2+3x+2x2, =[(x+1)2]2+3x(x+1)2+2x2, =[(x+1)2+2x][(x+1)2+x], =(x2+4x+1)(x2+3x+1); (3)(x+y)3+2xy(1﹣x﹣y)﹣1 =[(x+y)3﹣1]+2xy(1﹣x﹣y) =(x+y﹣1)[(x+y)2+x+y+1]﹣2xy(x+y﹣1) =(x+y﹣1)(x2+y2+x+y+1); (4)(x+3)(x2﹣1)(x+5)﹣20, =(x+3)(x+1)(x﹣1)(x+5)﹣20, =(x2+4x+3)(x2+4x﹣5)﹣20, =(x2+4x)2﹣2(x2+4x)﹣15﹣20, =(x2+4x+5)(x2+4x﹣7). 点评:此题主要考查分组分解法分解因式,综合利用了十字相乘法、公式法和提公因式法分解因式,难度较大,要灵活对待,还要有整体思想. |
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