Question

分解下列因式

(1)8a^3x^2-2a^5x^2
(2)a^2(x-y)+4y^2(y-x)
(3)4a^2b^2-(a^2+b^2)^2
(4)(x^2+4)^2-(2x+3)^2

Answer 1

(1)
8a^3x^2-2a^5x^2
= 2a^3x^2(4-a^2)
= 2a^3x^2(2+a)(2-a)

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

(3)
4a^2b^2-(a^2+b^2)^2
= (2ab+a^2+b^2)(2ab-a^2-b^2)
= -(a+b)^2(a-b)^2

(4)
(x^2+4)^2-(2x+3)^2
= (x^2+4+2x+3)(x^2+4-2x-3)
= (x^2+2x+7)(x^2-2x+1)
= (x^2+2x+7)(x^2-1)^2

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Answer 2

(1)8a^3x^2-2a^5x^2
=2a³x²(4-a²)
=2a³x²(2+a)(2-a)

(2)a^2(x-y)+4y^2(y-x)
=(x-y)(a²-4y²)
=(x-y)(a+2y)(a-2y)

(3)4a^2b^2-(a^2+b^2)^2
=(2ab+a²+b²)(2ab-a²-b²)
=-(a+b)²(a-b)²

(4)(x^2+4)^2-(2x+3)^2
=(x²+4+2x+3)(x²+4-2x-3)
=(x²+2x+7)(x²-2x+1)
=(x²+2x+7)(x-1)²