Tapeworm

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]

This problem origins from Mathematics for Engineer and Scientists 2nd ed - Alan Jeffrey P670 Chapter15.6 System of first order equations [latex]\displaystyle \dot{y}+2\begin{pmatrix} -3 & -2 \\ 5 & 3 \end{pmatrix} y = 5\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix} \begin{pmatrix} e^t \\ e^{-t} \end{pmatrix} [/latex] Try particular solution [latex]\displaystyle y_p=X\begin{pmatrix} e^t \\ e^{-t} \end{pmatrix}[/latex] [latex]\displaystyle \dot{y_p}=X\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix}e^t \\ e^{-t}\end{pmatrix}[/latex] substitute [latex]\displaystyle \dot{y_p}[/latex] and [latex]\displaystyle y_p[/latex] into the original diff eqt... [latex]\displaystyle X\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}+2\begin{pmatrix} -3 & -2 \\ 5 & 3 \end{pmatrix}X= 5\begin{pmatrix} 1 & -1 \\ -1 & 2 \end{pmatrix}[/latex] [latex]\displaystyle X=\begin{pmatrix} 3 & 3 \\ -5 & -4 \end{pmatrix}[/latex]

Yes, there are evidences, just my academic background is not enough to lay off them. Effort on my own is not enough to unveil the problem. Some of "them" are renowned professors, textbook authors and publishers. The allegation is serious? That's why I made sounds to call for public investigation.

Sorry, I mean I'm not knowledgeable to bring about the evidences, but still (have the right to) accuse "them".

I have a large grievance. Please listen to me, the facts may be embarrassing to educators, which I never thought of. We can even delete this thread when necessary. I am not knowledgeable to make an accusation. If you have a friend who is a practitioner or an engineer, he knows.

I have a perception that even the medium level science learning materials in English world are censored. Many books written in English are difficult to learn from. Even though there are numerous science & maths textbooks and internet materials, few of them are workable. Book comments in amazon often talk about this: poor notation, missing important ideas... I mean it is a kind of knowledge and technology prohibition to the public. By overwhelming publication of the fake textbooks to suffocate the public's engineering power. I don't know the situation in other countries(languages). Are foreign languages more transparent to science & technology?

forget it, the question is poor.

If there are handy of R3 vectors: r1, r2, r3, r4, r5, r6 Which 3 of them barely envelop a specified vector: a? One can change all:( r1, r2, r3, r4, r5, r6, a) into unit vectors and compare, but it is too clumsy...