Explainer: Identifying Polynomials

In this explainer, we will learn how to identify polynomial expressions.

In order to know if a function is a polynomial, we first need to know the definition of a polynomial.

Definition: Real Polynomial

An expression is a polynomial if it is of the form 𝑎+𝑎𝑥+𝑎𝑥++𝑎𝑥, where the coefficients 𝑎,𝑎,𝑎,,𝑎 are real constants and 𝑛 is a natural number.

This definition can be extended to polynomial functions.

Definition: Real Polynomial Function

A function, 𝑓(𝑥), is polynomial if it can be written in the form 𝑓(𝑥)=𝑎+𝑎𝑥+𝑎𝑥++𝑎𝑥, where the coefficients 𝑎,𝑎,𝑎,,𝑎 are real constants and 𝑛 is a natural number.

Now, we can identify polynomials and polynomial functions, using this definition, in the following example.

Example 1: Identifying Which Function Is a Polynomial

Which of the following functions is a polynomial?

  1. 𝑓(𝑥)=𝑥+𝑥+17
  2. 𝑓(𝑥)=𝑥+17𝑥
  3. 𝑓(𝑥)=𝑥+17
  4. 𝑓(𝑥)=18
  5. 𝑓(𝑥)=𝑥𝑥+𝑥17

Answer

Considering option A, the second term in the function is 𝑥, which we can also write as 𝑥. Using the definition of a polynomial, all the powers of 𝑥 need to have nonnegative whole exponents, and therefore A is not the correct answer.

For option B, the second term in the function is 17𝑥=17𝑥. The exponent of this power is negative so this function is not a polynomial.

In option C, one of the terms is 𝑥=𝑥. This function is not a polynomial since the exponent of this power is not an integer.

Option D is the function 𝑓(𝑥)=18. This is equivalent to 𝑓(𝑥)=18𝑥. Since the exponent of the only power of 𝑥 here is a nonnegative integer, this function is a polynomial.

We can check option E by expanding the brackets in the function to obtain 𝑓(𝑥)=𝑥+𝑥17𝑥.

Then, one of the terms has an exponent of 1 which is negative, so this is not a polynomial.

Now that we know how to identify if a function is a polynomial, let us consider the definition of the degree of a polynomial. We will build up to this from the definition of a monomial.

A monomial is a polynomial with only one term, for example, 3𝑥5𝑥.or

Definition: Degree of a Monomial

The degree of a monomial is the value of the exponent of the variable

A polynomial, as the name would suggest, is simply a sum of monomials. Using this definition for the degree of a monomial, we define the degree of a polynomial as follows.

Definition: Degree of a Polynomial

The degree of a polynomial is the highest degree of its monomials.

Let us look at some examples of how to find the degree of polynomials.

Example 2: Finding the Degree of a Polynomial

What is the degree of the polynomial 𝑥+6𝑥2𝑥+𝑥2?

Answer

We need to identify the term with the largest exponent. This is the term 𝑥, and the value of this exponent is 4; therefore, the degree of this polynomial is 4.

Example 3: Finding the Degree of a Simple Polynomial Function

What degree of polynomial is the function 𝑓(𝑥)=5𝑥+1?

Answer

Here, the term with the largest exponent is 5𝑥. We can also write this as 5𝑥, meaning that the value of its exponent is 1. Therefore, the degree of this polynomial is 1.

Example 4: Finding the Degree of a Polynomial Function

What degree of polynomial is the function 𝑓(𝑥)=7𝑥(3𝑥5)?

Answer

We start by expanding the brackets in the function to obtain 𝑓(𝑥)=7𝑥3𝑥+5. Next, we see that the term with the largest exponent is 3𝑥, and this exponent is 3. Therefore, the degree of this function is 3.

We will now look at some techniques which can enable us to find the degree of a polynomial function without having to perform difficult expansions.

How to Find the Degree of a Polynomial without Expanding

If we are given a polynomial of the form 𝑎+𝑎𝑥+𝑎𝑥++𝑎𝑥, and we are asked to find the degree of the polynomial function, 𝑓(𝑥)=𝑎+𝑎𝑥+𝑎𝑥++𝑎𝑥, then we could find its degree by expanding. However, this would be a difficult calculation to perform.

Alternatively, we could find the degree of this function by considering what the largest exponent from the resulting expansion would be. The term with the largest exponent will be (𝑎𝑥). Using an exponent rule, we get that this is equivalent to 𝑎𝑥().

Therefore, the degree of 𝑓(𝑥) will be 𝑛+𝑚.

Example 5: Finding the Degree of a Polynomial Function without Expanding Fully

Find the degree of the function 𝑓(𝑥)=3𝑥(6𝑥+10).

Answer

First, recall that when multiplying powers, you add exponents: 𝑥𝑥=𝑥.

When you raise a power to an exponent, you multiply the exponents: (𝑥)=𝑥.

If we were to expand (6𝑥+10), the power with the largest exponent would be (6𝑥). Using this, along with the exponent rules above, we have that the degree of 𝑓(𝑥)=3𝑥(6𝑥+10) is equal to the degree of 𝑔(𝑥)=𝑥(𝑥)=𝑥𝑥=𝑥.

Clearly the degree of 𝑔 is 4; therefore, the degree of 𝑓 is also 4.

To finish, let us recap some key points.

Key Points

  1. A polynomial is of the form 𝑎+𝑎𝑥+𝑎𝑥++𝑎𝑥.
  2. The degree of a monomial is the value of the exponent of the variable.
  3. A polynomial is a sum of monomials.
  4. The degree of a polynomial is the highest degree of its monomials.
  5. When finding the degree of complicated polynomials, we do not need to expand all the brackets. Instead, consider what the highest exponent would be as a result of the expansion.

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