Question

因式分解：x^2+4y^2+9z^2-4xy-12yz+6xz

(1)x^2+4y^2+9z^2-4xy-12yz+6xz
(2)x^3+6x^2+11x+6
(3)x^3+3x^2+3x-26
(4)x^3+4x^2-13x-24
(5)6x^4+5x^3+3x^2-3x-2
(6)2x^3-ax^2-5a^2x-2a^3

Answer 1

(1)x^2+4y^2+9z^2-4xy-12yz+6xz
=x^2-4xy+4y^2+6xz-12yz+9z^2
=(x-2y)^2+6z(x-2y)+9z^2
=(x-2y+3z)^2

(2)x^3+6x^2+11x+6
=x^3+x^2+5x^2+5x+6x+6
=x^2(x+1)+5x(x+1)+6(x+1)
=(x^2+5x+6)(x+1)
=(x+2)(x+3)(x+1)

(3)x^3+3x^2+3x-26
= x^3+3x^2+3x+1-27
=(x+1)^3-3^3
=(x+1-3)((x+1)^2+3(x+1)+3^2)
=(x-2)(x^2+5x+13)

(4)x^3+4x^2-13x-24
=(x^3-27)+(4x^2-13x+3)
=(x-3)(x^2+3x+9)+(4x-1)(x-3)
=(x-3)(x^2+3x+9+4x-1)
=(x-3)(x^2+7x+8)

(5)6x^4+5x^3+3x^2-3x-2
=6x^4+5x^3+x^2+2x^2-3x-2
=x^2(2x+1)(3x+1)+(2x+1)(x-2)
=(2x+1)(3x^3+x^2+x-2)
=(2x+1)(3x^3-3+x^2+x+1)
=(2x+1)[3(x-1)(x^2+x+1)+x^2+x+1]
=(2x+1)(3x-2)(x^2+x+1)

(6)2x^3-ax^2-5a^2x-2a^3
=2x^3-ax^2-(2a^2x+3a^2x)-2a^3
=(2x^3-2a^2x)-(ax^2+3a^2x+2a^3)
=2x(x+a)(x-a)-a(x+a)(x+2a)
=(x+a)[2x(x-a)-a(x+2a)]
=(x+a)[2x^2-3ax-2a^2]
=(x+a)(x-2a)(2x+a)

Answer 2

(1)x^2+4y^2+9z^2-4xy-12yz+6xz
=x^2-4xy+4y^2+6xz-12yz+9z^2
=(x-2y)^2+6z(x-2y)+9z^2
=(x-2y+3z)^2
(2)x^3+6x^2+11x+6
=x^3+x^2+5x^2+5x+6x+6
=x^2(x+1)+5x(x+1)+6(x+1)
=(x+1)(x^2+5x+6)
=(x+1)(x+2)(x+3)
(3)x^3+3x^2+3x-26
=x^3-2x^2+5x^2-10x+13x-26
=x^2(x-2)+5x(x-2)+13(x-2)
=(x-2)(x^2+5x+13)
(4)x^3+4x^2-13x-24
=x^3-3x^2+7x^2-21x+8x-24
=x^2(x-3)+7x(x-3)+8(x-3)
=(x-3)(x^2+7x+8)
(5)6x^4+5x^3+3x^2-3x-2
=6x^4-4x^3+9x^3-6x^2+9x^2-6x+3x-2
=2x^3(3x-2)+3x^2(3x-2)+3x(3x-2)+(3x-2)
=(3x-2)(2x^3+3x^2+3x+1)
=(3x-2)(2x^3+x^2+2x^2+x+2x+1)
=(3x-2)[x^2(2x+1)+x(2x+1)+(2x+1)]
=(3x-2)(2x+1)(x^2+x+1)
(6)2x^3-ax^2-5a^2x-2a^3
=2x^3+2ax^2-3ax^2-3a^2x-2a^2x-2a^3
=2x^2(x+a)-3ax(x+a)-2a^2(x-a)
=(x+a)(2x^2-3ax-2a^2)
=(x+a)(2x^2-4ax-ax-2a^2)
=(x+a)[2x(x-2a)-a(x-2a)]
=(x+a)(x-2a)(2x-a)

Answer 3

(1)x^2+4y^2+9z^2-4xy-12yz+6xz
=x^2-4xy+4y^2-12yz+6xz+9z^2
=(x-2y)^2+6z(x-2y)+9z^2
=(x-2y+3z)^2
(2)x^3+6x^2+11x+6
=x^3+x^2+5x^2+11x+6
=x^2(x+1)+(5x+6)(x+1)
=(x^2+5x+6)(x+1)
=(x+2)(x+3)(x+1)
(3)x^3+3x^2+3x-26
=x^3-8+3x^2+3x-18
=(x-2)(x^2+2x+4)+3(x^2+x-6)
=(x-2)(x^2+2x+4)+3(x-2)(x+3)
=(x-2)(x^2+2x+4+3(x+3))
=(x-2)(x^2+5x+13)
(4)x^3+4x^2-13x-24
=(x^3-27)+(4x^2-13x+3)
=(x-3)(x^2+3x+9)+(4x-1)(x-3)
=(x-3)(x^2+3x+9+4x-1)
=(x-3)(x^2+7x+8)
(5)6x^4+5x^3+3x^2-3x-2
=6x^4+5x^3+x^2+2x^2-3x-2
=x^2(2x+1)(3x+1)+(2x+1)(x-2)
=(2x+1)(3x^3+x^2+x-2)
=(2x+1)(3x^3-3+x^2+x+1)
=(2x+1)[3(x-1)(x^2+x+1)+x^2+x+1]
=(2x+1)(3x-2)(x^2+x+1)
(6)2x^3-ax^2-5a^2x-2a^3
=2x^3-ax^2-(2a^2x+3a^2x)-2a^3
=(2x^3-2a^2x)-(ax^2+3a^2x+2a^3)
=2x(x+a)(x-a)-a(x+a)(x+2a)
=(x+a)[2x(x-a)-a(x+2a)]
=(x+a)[2x^2-3ax-2a^2]
=(x+a)(x-2a)(2x+a)