几道因式分解题1-64v^62x^3-4x^2+x4a^2-6a-b^2+3ba^2-b^2-b-1/4a^2-4ab+b^2+3b10x^2-8ax-11x+4a+327x^3-y^3-9x^2-3xy-y^2a^2-b^2-x^2+y^2-2ay+2bxm^2+n^2-m-n+2mn-30x^2-2x-4y+1+4y^2+4xy

Question

几道因式分解题
1-64v^6
2x^3-4x^2+x
4a^2-6a-b^2+3b
a^2-b^2-b-1/4
a^2-4ab+b^2+3b
10x^2-8ax-11x+4a+3
27x^3-y^3-9x^2-3xy-y^2
a^2-b^2-x^2+y^2-2ay+2bx
m^2+n^2-m-n+2mn-30
x^2-2x-4y+1+4y^2+4xy

Answer 1

1-64v^6
=(1+8v^3)(1-8v^3)
=(1+2v)(1-2v+4v^2)(1-2v)(1-2v+4v^2)
立方和公式：a^3+b^3=(a+b)(a^2-ab+b^2)
立方差公式：a^3-b^3=(a-b)(a^2+ab+b^2)
2x^3-4x^2+x
=x(2x^2-4x+1)
设2x^2-4x+1=0
x^2-2x+(1/2)=0
x=(2±√2)/2
2x^3-4x^2+x=0
=2x(x-2-√2)(x-2+√2)
4a^2-6a-b^2+3b
=4a^2-6a+(9/4)-(b^2-3b+(9/4))=(2a-(3/2))^2-(b-(3/2))^2
=(2a-(3/2)+b-(3/2))(2a-(3/2)-b+(3/2))
=(2a+b-3)(2a-b)
a^2-b^2-b-1/4
=a^2-(b^2+b+(1/4))
=a^2-(b+(1/2))^2
=(a-b-(1/2))(a+b+(1/2))
a^2-4ab+b^2+3b
=
10x^2-8ax-11x+4a+3
=(4x^2-8ax+4a)
27x^3-y^3-9x^2-3xy-y^2
=
a^2-b^2-x^2+y^2-2ay+2bx
=(a^2-2ay+y^2)-(b^2-2bx+x^2)
=(a-y)^2-(b-x)^2
=(a-y-b+x)(a-y+b-x)
m^2+n^2-m-n+2mn-30
=(m^2+2mn+n^2)-(m+n)-30
=(m+n)^2-(m+n)-30
=(m+n-6)(m+n+5)
x^2-2x-4y+1+4y^2+4xy
=(x^2+4xy+4y^2)-(2x+4y)+1
=(x+2y)^2-2(x+2y)+1
=(x+2y-1)^2