Question

因式分解a^3+3^a^2+3a+b^3+3b^2+3b+2

因式分解：2、a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)
3、x^3+3xy+y^3-1
4、(x+y+Z)^3+(3x-2y-3Z)^3-(4x-y-2Z)^3

Answer 1

1.
a^3+3a^2+3a+b^3+3b^2+3b+2
=a^3+3a^2+3a+1+b^3+3b^2+3b+1
=(a+1)^3+(b+1)^3
=(a+1+b+1)[(a+1)^2-(a+1)(b+1)+(b+1)^2]
=(a+b+2)(a^2+2a+1-ab-a-b-1+b^2+2b+1)
=(a+b+2)(a^2+b^2-ab+a+b+1)

2.
a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)
=a^4+2a^2c^2+c^4-4bd(a^2+c^2)+4b^2d^2-b^4-2b^2d^2-d^4+4ac(b^2+d^2)-4a^2c^2
=(a^2+c^2)^2-4bd(a^2+c^2)+(2bd)^2-[(b^2+d^2)^2-4ac(b^2+d^2)+(2ac)^2]
=[(a^2+c^2)-2bd]^2-[(b^2+d^2)-2ac]^2
[平方差公式]
=[(a^2+c^2-2bd)+(b^2+d^2-2ac)][(a^2+c^2-2bd)-(b^2+d^2-2ac)]
=(a^2-2ac+c^2+b^2+d^2-2bd)(a^2+2ac+c^2-b^2-d^2-2bd)
=(a^2-2ac+c^2+b^2+d^2-2bd)[(a+c)^2-(b+d)^2]
=(a^2-2ac+c^2+b^2+d^2-2bd)(a+c+b+d)(a+c-b-d)

3.
x^3+y^3+3xy-1
=(x+y)^3-3x^2y-3xy^2+3xy-1
=(x+y)^3-3xy(x+y-1)-1
=[(x+y)-1][(x+y)^2+(x+y)+1]-3xy(x+y-1)
=(x+y-1)(x^2+2xy+y^2+x+y+1-3xy)
=(x+y-1)(x^2-xy+y^2+x+y+1)

4.

(x+y+z)3+(3x-2y-3z)3-(4x-y-2z)3
=(4x-y-2z)[(x+y+z)^2-2(x+y+z)(3x-2y-3z)+(3x-2y-3z)^2]-(4x-y-2z)3
=(4x-y-2z)[(x+y+z)^2-2(x+y+z)(3x-2y-3z)+(3x-2y-3z)^2-(4x-y-2z)^2]
=(4x-y-2z)[(x+y+z)((x+y+z)-(3x-2y-3z))+((3x-2y-3z)-(4x-y-2z))((3x-2y-3z)+(4x-y-2z))]
=(4x-y-2z)[(x+y+z)(-2x+3y+4z)+(-x-y-z)(7x-3y-5z)]
=(4x-y-2z)(x+y+z)[(-2x+3y+4z)-(7x-3y-5z)]
=(4x-y-2z)(x+y+z)(-9x+6y+9z)
=3(4x-y-2z)(x+y+z)(-3x+2y+3z)

太复杂了,应该加分啊.呵呵.

Answer 2

1.
a^3+3a^2+3a+b^3+3b^2+3b+2
=a^3+3a^2+3a+1+b^3+3b^2+3b+1
=(a+1)^3+(b+1)^3
=(a+1+b+1)[(a+1)^2-(a+1)(b+1)+(b+1)^2]
=(a+b+2)(a^2+2a+1-ab-a-b-1+b^2+2b+1)
=(a+b+2)(a^2+b^2-ab+a+b+1)

2.
a^4-b^4+c^4-d^4-2(a^2c^2-b^2d^2)+4ac(b^2+d^2)-4bd(a^2+c^2)
=a^4+2a^2c^2+c^4-4bd(a^2+c^2)+4b^2d^2-b^4-2b^2d^2-d^4+4ac(b^2+d^2)-4a^2c^2
=(a^2+c^2)^2-4bd(a^2+c^2)+(2bd)^2-[(b^2+d^2)^2-4ac(b^2+d^2)+(2ac)^2]
=[(a^2+c^2)-2bd]^2-[(b^2+d^2)-2ac]^2

=[(a^2+c^2-2bd)+(b^2+d^2-2ac)][(a^2+c^2-2bd)-(b^2+d^2-2ac)]
=(a^2-2ac+c^2+b^2+d^2-2bd)(a^2+2ac+c^2-b^2-d^2-2bd)
=(a^2-2ac+c^2+b^2+d^2-2bd)[(a+c)^2-(b+d)^2]
=(a^2-2ac+c^2+b^2+d^2-2bd)(a+c+b+d)(a+c-b-d)

3.
x^3+y^3+3xy-1
=(x+y)^3-3x^2y-3xy^2+3xy-1
=(x+y)^3-3xy(x+y-1)-1
=[(x+y)-1][(x+y)^2+(x+y)+1]-3xy(x+y-1)
=(x+y-1)(x^2+2xy+y^2+x+y+1-3xy)
=(x+y-1)(x^2-xy+y^2+x+y+1)

4.

(x+y+z)3+(3x-2y-3z)3-(4x-y-2z)3
=(4x-y-2z)[(x+y+z)^2-2(x+y+z)(3x-2y-3z)+(3x-2y-3z)^2]-(4x-y-2z)3
=(4x-y-2z)[(x+y+z)^2-2(x+y+z)(3x-2y-3z)+(3x-2y-3z)^2-(4x-y-2z)^2]
=(4x-y-2z)[(x+y+z)((x+y+z)-(3x-2y-3z))+((3x-2y-3z)-(4x-y-2z))((3x-2y-3z)+(4x-y-2z))]
=(4x-y-2z)[(x+y+z)(-2x+3y+4z)+(-x-y-z)(7x-3y-5z)]
=(4x-y-2z)(x+y+z)[(-2x+3y+4z)-(7x-3y-5z)]
=(4x-y-2z)(x+y+z)(-9x+6y+9z)
=3(4x-y-2z)(x+y+z)(-3x+2y+3z)

好难啊!

Answer 3

a^3+3a^2+3a+b^3+3b^2+3b+2
=a^3+3a^2+3a+1+b^3+3b^2+3b+1
=(a+1)^3+(b+1)^3
=(a+1+b+1)[(a+1)^2-(a+1)(b+1)+(b+1)^2]
=(a+b+2)(a^2+2a+1-ab-a-b-1+b^2+2b+1)
=(a+b+2)(a^2+b^2-ab+a+b+1)

Answer 4

a^3+3a^2+3a+b^3+3b^2+3b+2
=a^3+3a^2+3a+1+b^3+3b^2+3b+1
=(a+1)^3+(b+1)^3
=(a+1+b+1)[(a+1)^2-(a+1)(b+1)+(b+1)^2]
=(a+b+2)(a^2+2a+1-ab-a-b-1+b^2+2b+1)
=(a+b+2)(a^2+b^2-ab+a+b+1)