作业帮因式分解x^4-6x^2+1
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/29 12:11:41
因式分解x^4-6x^2+1
因式分解x^4-6x^2+1
因式分解x^4-6x^2+1
x^4 - 6x" + 1
= ( x" )" - 2x" + 1 - 4x"
= ( x" - 1 )" - ( 2x )"
= ( x" - 2x - 1 )( x" + 2x - 1 )
= ( x" - 2x + 1 - 2 )( x" + 2x + 1 - 2 )
= [ ( x - 1 )" - ( √2 )" ][ ( x + 1 )" - (...
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x^4 - 6x" + 1
= ( x" )" - 2x" + 1 - 4x"
= ( x" - 1 )" - ( 2x )"
= ( x" - 2x - 1 )( x" + 2x - 1 )
= ( x" - 2x + 1 - 2 )( x" + 2x + 1 - 2 )
= [ ( x - 1 )" - ( √2 )" ][ ( x + 1 )" - ( √2 )" ]
= ( x - 1 - √2 )( x - 1 + √2 )( x + 1 - √2 )( x + 1 + √2 )
其实，减少根号也完全有办法，
或许，被采纳的答案反而不对，
看一看，他的根号里面，也都是完全平方嘛，
3 + 2√2
= 2 + 2√2 + 1
= (√2)" + 2√2 + 1"
= ( √2 + 1 )"
3 - 2√2
= 2 - 2√2 + 1
= (√2)" - 2√2 + 1"
= ( √2 - 1 )"
当然，他这样配方
也的确是另一个方案
= ( x" )" - 6x" + 3" - 9 + 1
= ( x" - 3 )" - 8
= ( x" - 3 - 2√2 )( x" - 3 + 2√2 )
= [ x" - ( 3 + 2√2 ) ][ x" - ( 3 - 2√2 ) ]
可是我们就要看出，
括号里其实就是完全平方，
也就不需要加根号了
= [ x" - ( 2 + 2√2 + 1 ) ][ x" - ( 2 - 2√2 + 1 ) ]
= [ x" - ( √2 + 1 )" ][ x" - ( √2 - 1 )" ]
= [ x + ( √2 + 1 ) ][ x - ( √2 + 1 ) ][ x + ( √2 - 1 ) ][ x - ( √2 - 1 ) ]
= ( x + √2 + 1 )( x - √2 - 1 )( x + √2 - 1 )( x - √2 + 1 )
收起