Polynomial functions of degree three and higher

Basic functions: Polynomial functions

Polynomial functions of degree three and higher

Power functions, linear functions and quadratic functions are special cases of so-called polynomial functions. The general form of a polynomial function in the variable $x$ is:

$f(x)=a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0,$ with $a_n\neq 0$ .
The expression

$a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0,$ with $a_n\neq 0$ is called a polynomial. The parameters $a_0, a_1, \ldots, a_n$ are called the coefficients of the polynomial. The highest exponent $n$ is called the degree of the polynomial (hence linear and quadratic functions are also referred to as first degree functions and second degree functions, respectively). The term $a_nx^n$ called leading term of the polynomial (whereby the highest coefficient $a_n$ is not zero because otherwise you could have omitted this term).

The degree of $f(x)=4x^3-3x^2+1$ is 3; the leading term is $4x^3$ and the leading coefficient is $4$ .

The degree of $g(x)=\tfrac{1}{2}x^4-1$ is 4; the leading term is $\tfrac{1}{2}x^4$ and the leading coefficient is $\tfrac{1}{2}$ .

The degree of $h(x)= x^5-5x^3-x^2+4x+1$ is 5; the leading term is $x^5$ and the leading coefficient is $1$ .

Notation with summation symbol The polynomial

$a_nx^n+a_{n-1}x^{n-1} + \cdots + a_2x^2+a_1x+a_0$ has the following short notation via the summation symbol:

$\sum_{k=0}^{n}a_kx^k$

Power functions, linear and quadratic functions Linear and quadratic functions are polynomial functions of degree 1 and 2.
Power functions are polynomials functions with only one term.

You have alrteady seen that real quadratic functions can be factored as $f(x)=x^2-3x+2=(x-1)(x-2)$ . This also happens for higher degree polynomials. Expansion of brackets leads then to the standard form.

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New example

Graph The function

$f(x) = x^5-5x^3-x^2+4x+1$ whose graph over the interval $(-2.4, -2.4)$ is shown below is an example of a polynomial of degree 5 with leading term $x^5$ .

GeoGebra

JYou clearly see that the graph has 4 extrema and 5 zeros. In general it is true that the graph of a polynomial function of degree $n$ has $n-1$ extrema and $n$ zeros (at least when you ount with multiplicities, otherwise there can be less zeros).