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the Smallest Magnitude Root of a Polynomial Equation as the Constant Term Goes to Zero |
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Consider a polynomial equation in the standard form
Consider the smallest magnitude root of this equation as c_{0} → 0. That root would be proportional to c_{0}. For x very small the terms involving the higher powers of x become insignficant compared to c_{1}x and the polynomial equation reduces asymptotically to
Thus for small values of c_{0}, x=−c_{0}/c_{0} regardless of the magnitude of the other coefficients
The degree of the polynomial may be infinite as well as finite.
A function f(x) has a Maclaurin series of the form
Therefore the smallest root of f(x)=0 asymptotically approaches −f(0)/f'(0) as f(0)→0.
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