分解因式:(x⁴-4x²+1)(x⁴+3x²+1)+10x⁴

分解因式:(x⁴-4x²+1)(x⁴+3x²+1)+10x⁴
分解因式:(x⁴-4x²+1)(x⁴+3x²+1)+10x⁴

分解因式:(x⁴-4x²+1)(x⁴+3x²+1)+10x⁴
(x⁴-4x²+1)(x⁴+3x²+1)+10x⁴
=[(x⁴+1)-4x²][(x⁴+1)+3x²]+10x⁴
=(x⁴+1)²-x²(x⁴+1)-12x⁴+10x⁴
=(x⁴+1)²-x²(x⁴+1)-2x⁴
=(x⁴+1-2x²)(x⁴+1+x²)
=(x²-1)²[(x⁴+2x²+1)-x²]
=(x+1)²(x-1)²[(x²+1)²-x²]
=(x+1)²(x-1)²(x²+1+x)(x²+1-x)

原式=((x^4+1)-4x^2)((x^4+1)+4x^2-x^2)+10x^4
=((x^4+1)-4x^2)((x^4+1)+4x^2)-((x^4+1)-4x^2)x^2+10x^4
=(x^4+1)^2-16x^4-x^6+4x^4-x^2+10x^4
=x^8+2x^4+1-16x^4-x^6+4x^4-x^2+10x^4
...

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原式=((x^4+1)-4x^2)((x^4+1)+4x^2-x^2)+10x^4
=((x^4+1)-4x^2)((x^4+1)+4x^2)-((x^4+1)-4x^2)x^2+10x^4
=(x^4+1)^2-16x^4-x^6+4x^4-x^2+10x^4
=x^8+2x^4+1-16x^4-x^6+4x^4-x^2+10x^4
=x^8-x^6-x^2+1
=x^6(x^2-1)-(x^2-1)
=(x^6-1)(x^2-1)
=(x^2-1)(x^4+x^2+1)(x^2-1)
=(x+1)^2(x-1)^2(x^4+x^2+1)

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