Question

有几道因式分解不会，帮帮忙

16（6x-1)(2x-1)(3x+1)(x-1)+25
(6x-1)(2x-1)(3x-1)(x-1)+x的平方
（6x-1)(4x-1)(3x-1)(x-1)+9x的四次方
(2x-3y)的三次方+(3x-2y)的三次方-125(x-y)的三次方
x的四次方+x的三次方-4x的平方+x+1
4(2x^2-x+1)(x^2-2x+3)-(3x^2-3x+4)^2

Answer 1

16（6x-1)(2x-1)(3x+1)(x-1)+25
=16[（6x-1)(2x-1)][(3x+1)(x-1)]+25
=16(12x²-8x+1)(3x²-2x-1)+25
=4(12x²-8x+1)(12x²-8x-4)+25
设12x²-8x=a
原式
=4(a+1)(a-4)+25
=4(a²-3a-4)+25
=4a²-12a-16+25
=4a²-12a+9
=(2a-3)²
=(24x²-16x-3)²

(6x-1)(2x-1)(3x-1)(x-1)+x的平方
= [(6x-1)(x-1)][(2x-1)(3x-1)] + x²
= (6x²-7x+1)(6x²-5x+1) + x²
设6x²-6x+1=a
原式
=(a+x)(a-x)+x²
=a²-x²+x²
=a²
=(6x²-6x+1)²

（6x-1)(4x-1)(3x-1)(x-1)+9x的四次方
=[（6x-1)(x-1)][(4x-1)(3x-1)]+9x的四次方
=(6x²-7x+1)(12x²-7x+1)+9x^4
设9x²-7x+1=a
原式
=(a-3x²)(a+3x²)+9x^4
=a²-9x^4+9x^4
=a²
=(9x²-7x+1)²

(2x-3y)的三次方+(3x-2y)的三次方-125(x-y)的三次方
设2x-3y=a，3x-2y=b
则 a+b=5x-5y=5(x-y)
所以 125(x-y)³=[5(x-y)]³=(a+b)³
原式
=a³+b³-(a+b)³
=(a+b)³-3ab(a+b)-(a+b)³
=-3ab(a+b)
=-15(2x-3y)(3x-2y)(x-y)

x的四次方+x的三次方-4x的平方+x+1
=x^4-x³+2x³-2x²-2x²+2x-x+1
=x³(x-1)+2x²(x-1)-2x(x-1)-(x-1)
=(x-1)(x³+2x²-2x-1)
=(x-1)(x³-x²+3x²-3x+x-1)
=(x-1)[x²(x-1)+3x(x-1)+(x-1)]
=(x-1)²(x²+3x+1)

Answer 2

(6x-1)(2x-1)(3x-1)(x-1) + x²
= [(6x-1)(x-1)][(2x-1)(3x-1)] + x²
= (6x²-7x+1)(6x²-5x+1) + x²
= (6x²-6x+1 - x)(6x²-6x+1 + x) + x²
= (6x²-6x+1)² - x² + x²
= (6x² - 6x + 1)²