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???steps to factorize: a^2-ab+b^2-bc+c^2-ca


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I don't know the procedure to factorize:


[latex]a^2-ab+b^2-bc+c^2-ca[/latex]


into


[latex](a+\omega b+\omega^2 c)(a+\omega^2 b+\omega c)[/latex]


[latex]\omega[/latex] is complex cube root of unity: [latex]\omega^3=1[/latex]


=============================================

Can all the quadratic forms be factorized with complex roots?


[latex]ax^2+2fxy+by^2+2gyz+2px+2qy+d=0[/latex]
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I don't know the procedure to factorize:
[latex]a^2-ab+b^2-bc+c^2-ca[/latex]
into
[latex](a+\omega b+\omega^2 c)(a+\omega^2 b+\omega c)[/latex]
[latex]\omega[/latex] is complex cube root of unity: [latex]\omega^3=1[/latex]
=============================================
Can all the quadratic forms be factorized with complex roots?
[latex]ax^2+2fxy+by^2+2gyz+2px+2qy+d=0[/latex]

 

The trick is ω + ω² + 1 = 0. This gives the coefficients for ab, bc, and ac.

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