Monomials, Binomials, Polynomials

One of the most exciting parts of learning about algebra is expanding our vocabulary with the unique nomenclature used. Three of the most important words to learn are monomial, binomial, and polynomial. In this article, we'll define all three to deepen our understanding of mathematics. Let's begin!

Monomials, binomials, and polynomials: the monomials

A monomial can be defined as any product of numbers and variables. Examples include constants (like 17), variables with coefficients (like $3xy$ ), variables with exponents (like $-4x^2$ ), and variables with multiple exponents (like $a^5{b}^6{c}^7{d}^8{z}^{999}$ ). The examples adhere to the rules that variables are raised to positive integer powers and there are no plus or minus signs within the monomials.

The numerical part of a monomial is called the coefficient, and we can think of it as the non-variable part. For example, 3 is the coefficient of $3xy$ while x and y are the variables. Some monomials don't have obvious coefficients or variables, but they still qualify as monomials. For instance, 17 is a monomial even though it lacks a variable, while $x^6$ has a coefficient of one.

Monomials, binomials, and polynomials: the binomials

A binomial can be expressed as the sum of any two monomials. Examples include $x+3$ and $55x^2+33y^3$ . Likewise, remember that subtracting terms can be expressed using addition:

$x-3=x+(-3)$

This means that subtraction qualifies as binomials even though our definition includes the word "sum". If we're adding three monomials instead of 2, the result is called a trinomial. For example:

$5x-2y+8$

Monomials, binomials, and polynomials: the polynomials

A polynomial may be defined as the sum of n monomials for some whole number n. Monomials, binomials, and trinomials all represent the sum of a whole number of monomials even if the number is just 1, so they're all special cases of polynomials. There is no limit to the number of terms a polynomial can have.

Polynomials also have degrees that influence how they behave. The degree of a monomial is calculated by adding all of the exponents of all of its variables. For instance, the degree of $6xy{z}^4$ is 6 because we have $x_1,y_1$ , and $z^4$ . Most variables aren't labeled as being raised to the first power, so don't forget to include the 1s. Constants have no variables and therefore no exponents, so a monomial such as 23 has a degree of zero.

The degree of a polynomial is determined by the highest degree among its terms. For example, $x^2+1$ has a degree of 2 because 2 is the highest degree among its monomial terms.

Practice questions on monomials, binomials, and polynomials

a. Is this a monomial, binomial, or polynomial?

$4x-7$

It's best to approach problems like this by checking whether it's a monomial, binomial, or polynomial individually. There's a minus sign, which violates one of the rules monomials follow. Therefore, this is not a monomial. However, the minus sign separates two monomials, meaning that we are looking at a binomial. That also means we're looking at a polynomial because all binomials are special cases of polynomials. So, the answer is that this is a binomial and a polynomial but not a monomial.

b. What is the degree of this monomial: $a^{2}b{c}^5$

We need to add up all of the exponents to determine the degree of a monomial. The a is raised to the second power and the c to the 5th power, so you might be tempted to say $2+5=7$ and call it a day. However, remember that the b is raised to the first power as well. We need to add one more for an answer of eighth degree.

Topics related to the Monomials, Binomials, Polynomials

Degree (of a Monomial)

Comparing Linear, Polynomial, and Exponential Growth

Graphing Polynomial Functions

Flashcards covering the Monomials, Binomials, Polynomials

Algebra 1 Flashcards

College Algebra Flashcards

Practice tests covering the Monomials, Binomials, Polynomials

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College Algebra Diagnostic Tests

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