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the Smallest Magnitude
Root of a Polynomial
Equation as the Constant
Term Goes to Zero
Consider a polynomial equation in the standard form
Consider the smallest magnitude root of this equation as c0 → 0. That root would be proportional to c0. For x very small the terms involving the higher powers of x become insignficant compared to c1x and the polynomial equation reduces asymptotically to
Thus for small values of c0, x=−c0/c0 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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