Math Lesson 11.1.2 - The Definition of Polynomials

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Welcome to our Math lesson on The Definition of Polynomials, this is the second lesson of our suite of math lessons covering the topic of The Definition of Monomials and Polynomials, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Definition of Polynomials

"Mono" means "a single" in Greek. Thus, monomials are expressions that contain a single term where the coefficient and variables are multiplied by each other.

On the other hand, polynomials are algebraic expressions that contain two or more monomials that are added to or subtracted from each other. "Poly" means "many" in Greek. Hence, the term "polynomial" is translated as "many monomials put together".

For example,

is a polynomial, as it is made by three different monomials, 3x², 4x and -y (or -1y) that are connected to each other through the addition and subtraction symbols.

On the other hand,

is not a polynomial (it is an algebraic fraction, as explained in chapter 6), because the two terms are connected with each other through the division operation.

Likewise, the expression

is not a polynomial because not all its component terms are monomials (the second term contains a root, which is a rational power).

In this way, we reach a very important conclusion about the polynomials:

Polynomials are special algebraic expressions where all exponents of the variables are in whole powers (natural plus zero). In other words, polynomials belong to the algebraic expression set, but they do not represent the entire set of the algebraic expressions.

We express the general form of a polynomial as P(x, y, z, ⋯) where z, y, z, etc., are its variables. For example, it is clear

is a polynomial with three terms and two variables.

Monomials, therefore, represent a special case of polynomials; we can call them polynomials with one term.

Remark! If we have a number not followed by a variable in a polynomial, it still represents a term, but the number itself is not called a coefficient anymore but a constant instead.

Example 2

Identify the number of terms, coefficients, constants and variables of the following polynomial

1. -2x⁵ + 3y + 4
2. 4a³ b - 3a + b²

Solution 2

1. This polynomial has three terms: -2x⁵, 3y and 4. It also has two variables: x and y. The coefficients are -2 and 3, while the constant is 4.
2. This polynomial has three terms: 4a³b, -3a and b². It also has two variables: a and b. The coefficients are 4 and -3, while the constant is 0.

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4. Polynomials Revision Notes: The Definition of Monomials and Polynomials. Print the notes so you can revise the key points covered in the math tutorial for The Definition of Monomials and Polynomials
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6. Check your calculations for Polynomials questions with our excellent Polynomials calculators which contain full equations and calculations clearly displayed line by line. See the Polynomials Calculators by iCalculator™ below.
7. Continuing learning polynomials - read our next math tutorial: Operations with Polynomials

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