Standard Form of a Polynomial Function and Its Degree

Polynomials are a certain type of function, with very nice properties. Generally, a polynomial of a variable $x$ is of the form $$P(x) = \sum_{i=0}^{n} a_ix^i = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ for some constants $a_0$ through $a_n$.

Let's investigate some terminology. Definitions

The degree of a polynomial $P(x)$ is the highest power of $x$ that's non-zero (see below for this in more detail); so $$P(x) = \sum_{i=0}^{n} a_ix^i$$ has degree $n$, assuming of course that $a_n \neq 0$.

The standard form of a polynomial function is: $y = a_n x^{n} + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + ... + a_0 x^{0}$.

Let's try some examples.

Here comes a hard one. Get ready!

Polynomial functions are very interesting! So is their end behavior!

Polynomial Functions

When dealing with polynomial functions, it is easiest to deal with them in standard form. The degree of a polynomial determines the total number of roots, as stated by the fundamental theorem of algebra.

Standard form is when the terms of a mathematical expression are arranged such that the exponential values are in descending order. That means that the highest exponent (or highest degree) is listed first. By working with polynomials in standard form (and when we write the terms in this order), we can quickly and more efficiently do math!

The degree of a polynomial is the value of the highest power.

Looking Further

Once you've mastered the basics of the polynomial function, we can dive into operations on polynomial functions and finding the degree.

(Video) How to Write a Polynomial in Standard Form

by Mr. McLogan's Math Channel

In this video by Mr. McLogan's Math Channel he shows how to write any polynomial in its standard form.

(Video) Learn How to Write a Polynomial in Standard Form and Classify It

by Mr. McLogan's Math Channel

In this video by Mr. McLogan's Math Channel he works out a more complicated example of putting a polynomial in standard form, and then categorizes the polynomial.