A Key Limitation to Factor Theorem

I don’t really remember my Factor Theorem from John Abbott College well. But one detail which I think I do recall, is that its use was meant for “Univariate Polynomials”, with “Invariant Coefficients”. This means, that the coefficients needed to be integers or ‘other numbers’, known in advance, but not symbolic constants. In computerized cases where the coefficients aren’t preset, there are other, narrow constraints on them. A similar problem exists with the way I was taught to invert certain matrices in Linear Algebra. The elements are well-behaved in certain cases, but just as with polynomials, if the coefficients are suddenly random, floating-point numbers, those methods no longer work. Then, we must use a brute-force approach. And in the case of polynomials, there is no sure brute-force approach that works beyond the 4th degree.

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2 thoughts on “A Key Limitation to Factor Theorem”

  1. I suppose that to a computer, any numeric value can equally be a variable, while more formally, there is a difference between those and constants. Further, a computer can in fact be given an (n)th degree polynomial each coefficient of which is a floating-point number with finite precision, say 6 decimals. The computer can use successive approximation to find one root with the same precision, and then polynomial division to find a simpler polynomial of (n-1) degree.
    This can lead to an optimized, numerical approach to finding roots of polynomials of degrees much higher than 4.
    But the main issue I’d have with this, is that a numerical approach can only be useful in certain cases. In others, exact values are needed, such as the symbolic constants π or ℎ , in order to prove theorems or perform symbolic Algebra… And then Factor Theorem will leave us high and dry.

    1. The theory stands, that if (a) is a root of the polynomial, then to divide it by exactly (x-a) should leave a remainder of zero. But if we’ve allowed for some epsilon in the root, then the remainder will also be non-zero. In principle, we’d discard these remainders.
      However, one computational problem which may render the use of polynomials of greater degree than 8 faulty, is the fact that an error in the supposed root, will also snowball in producing a remainder each time.

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