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The Definition of a Polynomial in One Real Variable

08 March 2014 By Gideon In Polynomials
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Definition:


A  polynomial function of a real variable \(x\), is a function defined by the sum:
\[ P(x) = \sum_{k=0}^{n}c_k x^{k} = c_0 + c_1 x + c_2 x^{2}... + c_n x^{n}. \]

Each power of \(x\), e.g., \(x^{1}, x^{2}, x^{n}\), etc. is called monomial in \(x\).

A monomial times a constant coefficient, e.g., \(3x^{2}\) is called a term of the polynomial.

**WARNING**

This terminology is not entirely uniform; in particular some books define a ``monomial'' to be what we have defined to be a ``term,'' and vice-versa, and some make no distinction. 

Definition:

The degree, \(N,\) of a polynomial \(P(x)\), is the degree of the highest order term such that \(c_N x^{N} \neq 0\).

Example:


A polynomial of degree zero has the equation

\[P(x) = c_0. \]

I.e. it is a constant function.

Example:


A polynomial of degree one has the equation

\[ P(x) = c_0 + c_1x.\]

It is the equation of a straight line with:

  • \(y\)-intercept \(c_0\),
  • slope \(c_1\),
  • and \(x\)-intercept \(= \alpha = - \frac{c_0}{c_1}.\)

Example Problem:


Are there any polynomials in \(x\) that are also linear functions of \(x\)?

Solution:

The only polynomial in \(x\) that is also a linear function of \(x\) is the monomial \(P(x) = c_1x\).



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