[挑戰題]因式分解難題(B)

1) x^2+y^3+xy^2+2xy-1

2) x^4-4y^4-4xy^3-(xy)^2

3) (xy+2)^2 - (x-y)^2 -4xy

4) (2x-y)(x+y)-3y-2

5) a^4+a^2-2ab-b^2+1

引用來源連結

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(1)x^2+y^3+xy^2+2xy-1
=x^2+2xy+y^2+y^3+xy^2-y^2-1
=(x+y)^2+y^2(x+y)-(y^2+1)
=(x+y+y^2+1)(x+y-1) ~~~十字交乘

(2)x^4-4y^4-4xy^3-(xy)^2
=x^4-x^2y^2-4y^4-4xy^3
=x^2(x^2-y^2)-4y^3(y+x)
=x^2(x+y)(x-y)-4y^3(y+x)
=(x+y)[x^2(x-y)-4y^3]
=(x+y)(x^3-yx^2-4y^3)

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

4) (2x-y)(x+y)-3y-2=(2x-y-2)(x+y+1)
(2x-y)    -2
(x+y) +1

(5)a^4+a^2-2ab-b^2+1
=a^4+a^2-(2ab+b^2)+1
=a^4+a^2-(a^2+2ab+b^2)+1+a^2 (減a^2加a^2)
=a^4+2a^2+1-(a+b)^2
=(a^2+1)^2-(a+b)^2
=(a^2+1+a+b)(a^2+1-a-b)
=(a^2+a+b+1)(a^2-a-b+1)