factorization問題 maths高手入

1. a^6-1
=(a^3-1) (a^3+1)
=(a-1)(a^2 + a+1) (a+1)(a^2 - a +1)

2. x^4-10x^2+9
=(x^2 - 9)(x^2-1)
=(x-3)(x+3)(x-1)(x+1)

3. (a+b)^3+(a-b)^3
= [(a+b) + (a-b)] [(a+b)^2 - (a-b)(a+b) + (a-b)^2]
= (2a) {[(a+b) - (a-b) ]^2 + (a-b)(a+b)]
= 2a (4b^2 + a^2 - b^2)
=2a(3b^2 + a^2)

4.(a) x^4+2x^2y^2+y^4
=(x^2+y^2)(x^2-y^2)
=(x^2+y^2)(x-y)(x+y)

4.(b) x^4+x^2y^2+y^4
= x^2 + 2 x^2 y^2 + y^4 - x^2 y^2
= (x^2+y^2)^2 - x^2y^2
= [(x^2+y^2) - xy] [(x^2+y^2) +xy]
= (x^2 - xy +y^2) (x^2 + xy +y^2)

5(a) m^2+6m+9
=(m+3)^2

5(b) x^2+2xy+y^2+6(x+y)+9
= (x+y)^2 + 6(x+y) + 9
= [(x+y) +3]^2
=(x+y+3)^2

1.
a^6-1
=(a^3-1)(a^3+1)
=(a-1)(a^2+a+1)(a+1)(a^2-a+1)

2.
x^4-10x^2+9
=(x^2-9)(x^2-1)
=(x-3)(x+3)(x-1)(x+1)

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

4.
(a)
x^4+2x^2y^2+y^4
=(x^2+y^2)^2

4.
(b)
x^4+x^2y^2+y^4
=x^4+2x^2y^2+y^4-x^2y^2
=(x^2+y^2)^2 - x^2y^2
=(x^2+y^2+xy)(x^2+y^2-xy)

5.
(a)
m^2+6m+9
=(m+3)^2

5
(b)
x^2+2xy+y^2+6(x+y)+9
=(x+y)^2+6(x+y)+9
=[(x+y)+3]^2
=(x+y+3)^2

有錯請指正!
有問題歡迎提出!