因式分解(1)4x^2+4x-1(2)8x^6-27x^3y^6(3)5a-8a^2+42(4)3x^2-11x+6(5)√2m^2-(2-√3)m-√6(6)8x^2-6xy-35y^2(7)2x^4+5x^2-12(8)a^6-7a^3-8(9)(a^2+3a)^2-2(a^2+3a)-8(10)(x^2+8x+15)(x^2+8x+7)+15(11)x^4+4x^3-12x-9(12)x^3-y^3-x^2-xy-y^2(13)x^4-14x^2+1(14)

Question

因式分解
(1)4x^2+4x-1
(2)8x^6-27x^3y^6
(3)5a-8a^2+42
(4)3x^2-11x+6
(5)√2m^2-(2-√3)m-√6
(6)8x^2-6xy-35y^2
(7)2x^4+5x^2-12
(8)a^6-7a^3-8
(9)(a^2+3a)^2-2(a^2+3a)-8
(10)(x^2+8x+15)(x^2+8x+7)+15
(11)x^4+4x^3-12x-9
(12)x^3-y^3-x^2-xy-y^2
(13)x^4-14x^2+1
(14)a^8+a^4b^4+b^8

Answer 1

(1)4x^2+4x-1=4x^2+4x+1-2=(2x+1)^2-2
=(2x+1-√2)(2x+1+√2)
(2)8x^6-27x^3y^6=x^3(8x^3-27y^6)
=x^3(2x-3y^2)(4x^2+6xy^2+9y^4)
(3)5a-8a^2+42=-(8a^2-5a-42)
=-8[a-(5+8√21)/16][a-(5-8√21)/16]
(4)3x^2-11x+6=(3x-2)(x-3)
(5)√2m^2-(2-√3)m-√6=(√2m+√3)(m-√2)
(6)8x^2-6xy-35y^2=(4x+7y)(2x-5y)
(7)2x^4+5x^2-12=(2x^2-3)(x^2+4)
=(√2x+√3)(√2x-√3)(x^2+4)
(8)a^6-7a^3-8=(a^3+1)(a^3-8)
=(a+1)(a^2-a+1)(a-2)(a^2+2a+4)
(9)(a^2+3a)^2-2(a^2+3a)-8
=(a^2+3a-4)(a^2+3a+2)
=(a+4)(a-1)(a+2)(a+1)
(10)(x^2+8x+15)(x^2+8x+7)+15
=(x^2+8x+7)^2+8(x^2+8x+7)+15
=(x^2+8x+7+3)(x^2+8x+7+5)
=(x^2+8x+10)(x+2)(x+6)
(11)x^4+4x^3-12x-9
=(x^2+3)(x^2-3)+4x(x^2-3)
=(x^2-3)(x^2+4x+3)
=(x-√3)(x+√3)(x+3)(x+1)
(12)x^3-y^3-x^2-xy-y^2
=(x-y)(x^2+xy+y^2)-(x^2+xy+y^2)
=(x^2+xy+y^2)(x-y-1)
(13)x^4-14x^2+1
=x^4-14x^2+49-48
=(x^2-7)^2-48
=(x^2-7+4√3)(x^2-7-4√3)
(14)a^8+a^4b^4+b^8
=a^8+2a^4b^4+b^8-a^4b^4
=(a^4+b^4)^2-a^4b^4
=(a^4+a^2b^2+b^4)(a^4-a^2b^2+b^4)
=(a^2+b^2+ab)(a^2+b^2-ab)(a^2+b^2+√3ab)(a^2+b^2-√3ab)