Đáp án+Giải thích các bước giải:

 `1)x^{18}-2x^{9}+1`

`=(x^{9})^{2}-2.x^{9}.1+1^{2}`

`=(x^{9}-1)^{2}`

`=[(x^{3})^{3}-1^{3}]^{2}`

`={(x^{3}-1^{3}).[(x^{3})^{2}+x^{3}.1+1^{2}]}^{2}`

`=(x^{3}-1^{3})^{2}.(x^{6}+x^{3}+1)^{2}`

`=[(x-1).(x^{2}+x+1)]^{2}.(x^{6}+x^{3}+1)^{2}`

`=(x-1)^{2}.(x^{2}+x+1)^{2}.(x^{6}+x^{3}+1)^{2}`

`2)x^{18}+2x^{9}+1`

`=(x^{9})^{2}+2.x^{9}.1+1^{2}`

`=(x^{9}+1)^{2}`

`=[(x^{3})^{3}+1^{3}]^{2}`

`={(x^{3}+1^{3}).[(x^{3})^{2}-x^{3}.1+1^{2}]}^{2}`

`=(x^{3}+1^{3})^{2}.(x^{6}-x^{3}+1)^{2}`

`=[(x+1).(x^{2}-x+1)]^{2}.(x^{6}-x^{3}+1)^{2}`

`=(x+1)^{2}.(x^{2}-x+1)^{2}.(x^{6}-x^{3}+1)^{2}`

`3)x^{18}+5778x^{9}+1`

`=(x^{9})^{2}+2.x^{9}.2889+2889^{2}+1-2889^{2}`

`=(x^{9}+2889)^{2}+(1-2889).(1+2889)`

`=(x^{9}+2889)^{2}-2888.2890`

`=(x^{9}+2889)^{2}-(\sqrt{2888.2890})^{2}`

`=(x^{9}+2889-\sqrt{2888.2890}).(x^{9}+2889+\sqrt{2888.2890})`

`=[x^{9}+\sqrt{2889}.(\sqrt{2889}-\sqrt{2888})].[x^{9}+\sqrt{2889}.(\sqrt{2889}+\sqrt{2888})]`

`4)x^{18}+103x^{9}+512`

`=(x^{9})^{2}+2.x^{9}.\frac{103}{2}+(\frac{103}{2})^{2}+512-(\frac{103}{2})^{2}`

`=(x^{9}+\frac{103}{2})^{2}+\frac{2048}{4}-\frac{10609}{4}`

`=(x^{9}+\frac{103}{2})^{2}-\frac{8561}{4}`

`=(x^{9}+\frac{103}{2})^{2}-(\frac{\sqrt{8561}}{2})^{2}`

`=(x^{9}+\frac{103}{2}-\frac{\sqrt{8561}}{2}).(x^{9}+\frac{103}{2}+\frac{\sqrt{8561}}{2})`

`=(x^{9}+\frac{103-\sqrt{8561}}{2}).(x^{9}+\frac{103+\sqrt{8561}}{2})`