Essential Degree

The essential degree (their normal degree) is fairly straightforward to determine for a polynomial, but for root functions this takes a little more effort. For instance, the essential degree of √--2--- x + 1 is √ -2- x, which is x (for positive x) or -x (for negative x).

To determine the essential degree of √--2--- x + 1 above, we dropped all terms under the radical sign except that term whose exponent gave us the degree of the polynomial underneath the radical sign. We isolate this highest powered term because as x gets bigger while x →∞ , the term with the highest power of x dominates the others:

x = 10 1,000 1,000,000
x2 =100 1,000,000 1,000,000,000,000
x3 =10001,000,000,0001,000,000,000,000,000,000

or as x →-∞:

x = -10 -1,000 -1,000,000
x2 =100 1,000,000 1,000,000,000,000
x3 =-1000-1,000,000,000-1,000,000,000,000,000,000

so we see that when x gets larger and larger, the contributions by the terms with the smaller exponents will be negligible.

Similarly, the essential degree of √ ----------------- x4 + x3 + 9x2 + 1 is √ --- x4 is x2.

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