# MandysNotes

## The Definition of a Polynomial in One Real Variable

08 March 2014 By Gideon
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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 $$P(x) = c_1x$$.