如何分解下列因式1.（x^2-y^2-z^2)^2-4y^2z^22.(a-b)^3-(a-b)3.x^3+9x^2+26x+244.x^3-9x+85.(x+1)(x+3)(x+5)(x+7)+156.x^4+x^3-2x^2-x+1

Question

如何分解下列因式
1.（x^2-y^2-z^2)^2-4y^2z^2 2.(a-b)^3-(a-b) 3.x^3+9x^2+26x+24 4.x^3-9x+8 5.(x+1)(x+3)(x+5)(x+7)+15
6.x^4+x^3-2x^2-x+1

Answer 1

( x" - y" - z" )" - 4y"z"
= ( x" - y" - z" )" - ( 2yz )"
= ( x" - y" - z" - 2yz )( x" - y" - z" + 2yz )
= [ x" - ( y" + 2yz + z" ) ][ x" - ( y" - 2yz + z" ) ]
= [ x" - ( y + z )" ][ x" - ( y - z )" ]
= [ x + ( y + z ) ][ x - ( y + z ) ][ x + ( y - z ) ][ x - ( y - z ) ]
= ( x + y + z )( x - y - z )( x + y - z )( x - y + z )

( a - b )"' - ( a - b )
= ( a - b )[ ( a - b )" - 1 ]
= ( a - b )( a - b - 1 )( a - b + 1 )

x"' + 9x" + 26x + 24

x" - 9x + 8
= x" - x - 8x + 8
= x( x - 1 ) - 8( x - 1 )
= ( x - 1 )( x - 8 )

( x + 1 )( x + 3 )( x + 5 )( x + 7 ) + 15
= ( x + 7 )( x + 1 )( x + 3 )( x + 5 ) + 15
= ( x" + 8x + 7 )( x" + 8x + 15 ) + 15
= ( x" + 8x )" + 15( x" + 8x ) + 7( x" + 8x ) + 105 + 15
= ( x" + 8x )" + 22( x" + 8x ) + 120
= ( x" + 8x )" + 10( x" + 8x ) + 12( x" + 8x ) + 120
= ( x" + 8x )( x" + 8x + 10 ) + 12( x" + 8x + 10 )
= ( x" + 8x + 10 )( x" + 8x + 12 )
= ( x" + 8x + 10 )( x" + 2x + 6x + 12 )
= ( x" + 8x + 10 )[ x( x + 2 ) + 6( x + 2 ) ]
= ( x + 2 )( x + 6 )( x" + 8x + 10 )

x^4 + x"' - 2x" - x + 1
= x"( x" + x - 2 ) - ( x - 1 )
= x"( x" - x + 2x - 2 ) - ( x - 1 )
= x"[ x( x - 1 ) + 2( x - 1 ) ] - ( x - 1 )
= x"( x + 2 )( x - 1 ) - ( x - 1 )
= ( x - 1 )( x"' + 2x" - 1 )