San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Asymptotic Limit of
the Smallest Magnitude
Root of a Polynomial
Equation as the Constant
Term Goes to Zero

Consider a polynomial equation in the standard form

cnxn + cn-1xn-1 + + c1x + c0 = 0

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

c1x + c0 = 0
and thus
x = − c0/c1 = 0

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

f(x) = f(0) + f'(0)x + ½f"(0)x² + …

Therefore the smallest root of f(x)=0 asymptotically approaches −f(0)/f'(0) as f(0)→0.

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