x"' - x" - x - 2
= x"' - 1 - x" - x - 1
= ( x - 1 )( x" + x + 1 ) - ( x" + x + 1 )
= ( x" + x + 1 )( x - 1 - 1 )
= ( x - 2 )( x" + x + 1 )
或者
= x"' + x" + x - 2x" - 2x - 2
= x( x" + x + 1 ) - 2( x" + x + 1 )
= ( x - 2 )( x" + x + 1 )
x^4 - 6x" - 7x - 6
= x^4 - x - 6x" - 6x - 6
= x( x"' - 1 ) - 6( x" + x + 1 )
= x( x - 1 )( x" + x + 1 ) - 6( x" + x + 1 )
= ( x" + x + 1 )( x" - x - 6 )
= ( x" + x + 1 )( x" - 3x + 2x - 6 )
= ( x" + x + 1 )[ x( x - 3 ) + 2( x - 3 ) ]
= ( x - 3 )( x + 2 )( x" + x + 1 )
刚才的 x^8 + x + 1,这三个多项式
都有一个因式是 ( x" + x + 1 ),
分解因式,我们就要熟悉立方和、立方差,
a"' + b"' = ( a + b )( a" - ab + b" )
a"' - b"' = ( a - b )( a" + ab + b" )
a"' - 1 = ( a - 1 )( a" + a + 1 )
解题思路,可以先算一算,
2^8 + 2 + 1 = 256 + 3 = 259
分解因数,
259 = 210 + 49 = 7 X 37
看到因数 7,我们就能够想想立方差,
= ( 4 + 2 + 1 ) X 37 = ( 2" + 2 + 1 ) X 37
另一个因数 37 是质数,
我们还要看看因数 1,
别忘了可能还有因式 ( x - 1 ) = ( 2 - 1 ) = 1
积极思想,开动脑筋,
