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No closed form solution means we can't write out the solution cleanly using standard algebraic operations. You can still write iterative algorithms that converg

No closed form solution means we can't write out the solution cleanly using standard algebraic operations. You can still write iterative algorithms that converge on the solution, such as by solving for the eigenvalues of a matrix.

thaumasiotes 4y ago
Those won't tell you the solution; they'll tell you a number that is near the solution. When the solution is irrational, as is common for polynomials, it might be difficult to guess what it is based on knowing an approximation of it.
- suzumer 4y ago
  When using 64 bit floating point numbers with 54 bits of precision, (or likely in this case, complex values consisting of two 64 bit floats) and you use an algorithm which you can guarantee converges to within 55 bits of precision in X steps, you have found the closest representable solution. Yes, you don't know whether it's 1 + sqrt(17) or 0.9 + Sqrt[17.84...], but you know to a degree of precision that 95% of use cases need what the value is. I do understand what you are saying, but the parent comment was referring to a student factoring polynomials. You doubted what he did based on the fact that he couldn't exactly specify what the roots were. I just assumed that he likely found the roots to a reasonable accuracy and then factored with these approximated values. The parent never specified, but given doubting the authenticity of the comment or celebrating youthful ingenuity, I choose the latter. I understand what you are saying. Yes, you can not find the EXACT values of the roots, but within the context of this comment chain, I don't think that matters.
  - thaumasiotes 4y ago
    On the contrary, when the problem posed is "find the roots of this polynomial", it's very common to accept approximate values. When the problem posed is "factor this polynomial", it's unheard of. This is the conceptual difference between "needing a number to work with" and "studying math".

thaumasiotes 4y ago

Those won't tell you the solution; they'll tell you a number that is near the solution. When the solution is irrational, as is common for polynomials, it might be difficult to guess what it is based on knowing an approximation of it.

suzumer 4y ago
When using 64 bit floating point numbers with 54 bits of precision, (or likely in this case, complex values consisting of two 64 bit floats) and you use an algorithm which you can guarantee converges to within 55 bits of precision in X steps, you have found the closest representable solution. Yes, you don't know whether it's 1 + sqrt(17) or 0.9 + Sqrt[17.84...], but you know to a degree of precision that 95% of use cases need what the value is. I do understand what you are saying, but the parent comment was referring to a student factoring polynomials. You doubted what he did based on the fact that he couldn't exactly specify what the roots were. I just assumed that he likely found the roots to a reasonable accuracy and then factored with these approximated values. The parent never specified, but given doubting the authenticity of the comment or celebrating youthful ingenuity, I choose the latter. I understand what you are saying. Yes, you can not find the EXACT values of the roots, but within the context of this comment chain, I don't think that matters.
- thaumasiotes 4y ago
  On the contrary, when the problem posed is "find the roots of this polynomial", it's very common to accept approximate values. When the problem posed is "factor this polynomial", it's unheard of. This is the conceptual difference between "needing a number to work with" and "studying math".