1 Polynomials

A polynomial in $x$ is a function of the form

$p\left(x\right)=a_0+a_{1}x+a_2{x}^2+\dots a_n{x}^n$ ( $a_n\ne 0,n$ a non-negative integer)

where $a_0,a_1,a_2,\dots ,a_n$ are constants. We say that this polynomial $p$ has degree equal to $n$ . (The degree of a polynomial is the highest power to which the argument, here it is $x$ , is raised.) Such functions are relatively simple to deal with, for example they are easy to differentiate and integrate. In this Section we will show ways in which a function of interest can be approximated by a polynomial.

First we briefly ensure that we are certain what a polynomial is.

Example 1

Which of these functions are polynomials in $x$ ? In the case(s) where $f$ is a polynomial, give its degree.

1. $f\left(x\right)=x^2-2-\frac{1}{x}$ ,
2. $f\left(x\right)=x^4+x-6,$
3. $f\left(x\right)=1,$
4.   $f\left(x\right)=mx+c,$   $m$ and $c$ are constants.
5. $f\left(x\right)=1-x^6+3x^3-5x^3$

Solution

1. This is not a polynomial because of the $\frac{1}{x}$ term (no negative powers of the argument are

     allowed in polynomials).
   
   

2. This is a polynomial in $x$ of degree 4.
   
   
3. This is a polynomial of degree 0.
   
   
4. This straight line function is a polynomial in $x$ of degree 1 if $m\ne 0$ and of degree 0 if $m=0$ .
5. This is a polynomial in $x$ of degree 6.

Task!

Which of these functions are polynomials in $x$ ? In the case(s) where $f$ is a polynomial, give its degree.

1. $f\left(x\right)=\left(x-1\right)\left(x+3\right)$
2. $f\left(x\right)=1-x^7$
3. $f\left(x\right)=2+3e^x-4e^{2x}$
4. $f\left(x\right)=cos\left(x\right)+{sin}^2\left(x\right)$

Answer

We have in fact already seen, in HELM booklet  16, one way in which some functions may be approximated by polynomials. We review this next.