Question

Factor the expression completely over the complex numbers.

x^4 + 10x^2 + 25



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Answer 1

`\displaystyle\\x^4+10x^2+25=\\\\=x^4+5x^2+5x^2+25=\\\\=(x^4+5x^2)+(5x^2+25)=\\\\=x^2(x^2+5)+5(x^2+5)=\\\\=(x^2+5)(x^2+5)=\\\\=\left(x^2+\left(√(5)\right)^2\right)\left(x^2+\left(√(5)\right)^2\right)=\\\\=\Big(x+i√(5)\Big)\Big(x-i√(5)\Big)\Big(x+i√(5)\Big)\Big(x-i√(5)\Big)=\\\\=\Big(x+i√(5)\Big)\Big(x+i√(5)\Big)\Big(x-i√(5)\Big)\Big(x-i√(5)\Big)=\\\\=\boxed{\Big(x+i√(5)\Big)^2\Big(x-i√(5)\Big)^2}`