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Apr26

Review the Properties of Polynomials

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:

a2bc3 –> Degree =6
-3a3c6  –> Degree = 9
6ac4  –> Degree = 5
2b –> Degree = 1

Therefore, the degree of the polynomial P is 9.

The polynomial Q has two monomials:

-10p6  –> Degree = 6
pq2  –> Degree = 3

Therefore, the polynomial Q has a degree of 6.

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

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?
polynomials

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?

polynomials

With this we have finished the post for this week.

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