Explainer: Degree and Coefficient of Polynomials

In this explainer, we will learn how to determine the degree of a polynomial and use the terminology associated with polynomials, such as terms, coefficients, and constants.

Definition: Polynomial

A polynomial is an expression consisting of one or more monomials, where a monomial is defined as a product of constants and variables with nonnegative integer powers.

A few examples of expressions that are polynomials are the following:

  1. 𝑥+5𝑥
  2. 𝑥𝑦+5𝑥
  3. 𝑥
  4. 56𝑥𝑦𝑧+18𝑥5𝑧
  5. 12𝑥

Examples of expressions that are not polynomials are as follows:

  1. 𝑥
  2. 𝑥+3𝑥
  3. 𝑥+6𝑥
  4. 36𝑥𝑦𝑧+5𝑥4𝑥𝑦
  5. 6𝑥+8𝑦

One point that we should note, before looking further into the identification of polynomials, is that another name for a monomial is a term and that these two words are often used interchangeably.

We also need to define what we mean when we talk about the degree of a polynomial and what we mean when we talk about a coefficient.

Definition: Degree of a Polynomial

The degree of a polynomial is defined as the greatest sum of the exponents of the variables in any single term.

For example, if we consider the polynomial 5𝑥𝑦+4𝑥+3𝑥𝑦5𝑥, the sum of the exponents is greatest in the first term which has a sum of 5; therefore, the degree of the polynomial is 5.

Definition: Coefficient

The coefficient of a term is the number used to multiply any variables. This is generally the number at the start of an algebraic term. The coefficient is also defined as the numerical factor in an algebraic term.

For example, if we consider again the polynomial 5𝑥𝑦+4𝑥+3𝑥𝑦5𝑥, the coefficient of the first term is 5, the coefficient of the second term is 4, and so on. Let us have a look at some examples that ask about identifying polynomials and properties of polynomials.

Example 1: Identifying Polynomials

Which of the following expressions are polynomials?

  1. 𝑥+5𝑥𝑦2
  2. 𝑥𝑦
  3. 𝑥𝑦
  4. 5𝑥4𝑥𝑦

Answer

To determine which of the expressions are polynomials, we need to establish if they only contain monomials. Recall that a monomial is defined as a product of constants and variables with nonnegative integer powers. The first expression contains three monomials and is a polynomial, and the second expression is a monomial and, thus, a polynomial. However, the third expression contains a negative exponent and, therefore, is not a polynomial, nor is the fourth expression as 5𝑥 can be rewritten as 5𝑥 which contains a negative power and is not a monomial.

Example 2: Identifying the Degree of a Polynomial

Determine the degree of 𝑦7𝑦.

Answer

With this question, we have a polynomial that contains a single variable. When this is the case, the degree of the polynomial is defined as the variable with the greatest exponent. In this case, the first term has an exponent of 4, which means that the degree is 4.

Example 3: Identifying the Degree and Coefficients of a Polynomial

Determine the coefficient and degree of 7𝑥.

Answer

The coefficient is the numerical factor of an algebraic term. So, the coefficient of this term is 7. The degree of an algebraic term is the sum of the exponents of the algebraic factors. The term 7𝑥 has the degree 3 because its only exponent is 3.

Example 4: Identifying the Degree of a Polynomial

If the degree of 7𝑥 is the same as that of 6𝑦, what is the value of 𝑛?

Answer

Using the fact that the degree of an algebraic term is the sum of the exponents of the algebraic factors, we can see that the degree of 7𝑥 is 5 as it contains only one variable. Therefore, we know that the degree of 6𝑦 is 5, so the 𝑛 must be equal to 5.

We may also be asked about particular features of a polynomial, for example, how many terms does it have, or which of the monomials is the constant term. Let us look at a couple of examples of questions like this.

Example 5: Identifying the Number of Terms in a Polynomial

How many terms are in the following expression? 4𝑥𝑦+27

Answer

Recall that a term is a product of constants and variables with nonnegative integer powers. Each of the components of this expression are terms as they meet this criterion.

Additionally, the sum of one or more terms is a polynomial. In this question, we have a polynomial that contains three terms.

Example 6: Identifying the Constant Term of a Polynomial

Which is the constant term in the following expression? 4𝑥𝑦+27

Answer

Recall that a constant term is a term that has a value that does not vary; that is, it is always constant. The term 4𝑥 contains the variable 𝑥, the term 𝑦 contains the variable 𝑦, but the term 27 is fixed. This, therefore, is the constant term in the expression.

To finish, let us look at one more slightly trickier example that pulls together our understanding of the degree of a polynomial and whether an expression is a polynomial.

Example 7: Identifying Polynomials

Which of the following expressions are polynomials with degree 5?

  1. 𝑥+5𝑥2
  2. 𝑥𝑦𝑥3𝑥
  3. 𝑥𝑦4𝑥𝑦
  4. 𝑥𝑦𝑥
  5. 𝑥2𝑥𝑦+5𝑥

Answer

Our first step when answering this question is to establish if all of the expressions are in fact polynomials. If we look carefully at the five expressions, only expression (D) is not a polynomial as it contains a variable raised to a negative power. We have to be particularly careful here as the expression does contain a term raised to the power of 5 so it would be easy to make the mistake of identifying it as a polynomial of degree 5.

Having established that the other four expressions are all polynomials, we need to identify which of them have degree 5. Recall that the degree of an algebraic term is the sum of the exponents of the algebraic factors. Polynomial (A) only contains a single variable, so the degree is the variable that has the highest power, which in this case is 6.

If we sum the exponents of the variables of the first term of polynomial (B) 𝑥𝑦, we get a total of 5, so this polynomial has degree 5. Polynomial (C) is very similar as the sum of the exponents of its first term 𝑥𝑦 is 5. Finally, if we look at polynomial (E) the greatest sum of exponents happens to be the third term 5𝑥 and, therefore, this too has a degree of 5.

So, the polynomials with degree 5 are (B), (C), and (E).

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