In this post we will look at the properties of **polynomials**, what they are, what degree they have, and whether or not they complete and/or in standard form.

Let’s start by looking at** what polynomials are.**

**A polynomial is an assembly of monomials.
**

Polynomials are usually named with capital letters (usually starting with the letter P) and the polynomial variables are written in parentheses.

P (a, b, c) = a ^{2} bc ^{3} – 3rd ^{3} c ^{6} + 6ac ^{4} + 2b

The polynomial P contains the variables a, b and c. It is composed of 4 monomials, therefore we can say that the polynomial has** 4 terms.**

Q (p, q) = -10p ^{6} + pq ^{2}

The polynomial Q contains the variables p and q. It is composed of two monomials, therefore we can say that the polynomial has **2 terms.**

### Degree of a polynomial

The *degree* of a polynomial is the largest of the degrees of their monomials.

Let’s look at the degrees of the above polynomials:

The polynomial P has 4 monomials:

a^{2}bc^{3} –> Degree =6

-3a^{3}c^{6} –> Degree = 9

6ac^{4} –> Degree = 5

2b –> Degree = 1

Therefore, the **degree of the polynomial P is 9****.**

The polynomial Q has two monomials:

-10p^{6} –> Degree = 6

pq^{2} –> Degree = 3

Therefore, the **polynomial Q has a degree of 6****.**

*What do you think is the degree of this polynomial?*

### Polynomials in Standard Form

A polynomial is in *standard form *when its monomials are ordered from **largest to smallest degree.**

Returning to the previous polynomials, the polynomial P has monomials of degree 6, 9, 5, and 1. As they are not ordered from largest to smallest, we can say that the polynomial P is not in standard form*.*

The polynomial Q has the monomials of degrees: 6, 3. In this case the degrees of the monomials are ordered from largest to smallest degree, therefore the polynomial Q is in standard form.

*Is the following polynomial ordered?*

### Complete Polynomials

A polynomial is *complete* when you have all terms with degrees from the largest to the zero degree.

The polynomial P containing degrees 6, 9, 5 and 1 is incomplete, because terms of degree 8, 7, 4, 3, 2 and 1 are missing.

The Q polynomial is also incomplete because it has terms of degree 6 and 3, thus missing terms of degree 5, 4, 2, 1 and 0.

A final exercise: Is this polynomial complete?

With this we have finished the post for this week.

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Learn More:

- Learn How to Add Polynomials with Examples
- Representing Polynomials with Algebra Tiles
- How to Subtract Polynomials with Help from Algebra Tiles
- Learn the Properties of Monomials
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