What Is a Polynomial? Notation, Value & Other Related Concepts

In this tutorial, we’ll learn all about polynomials. We’ll look at what polynomials are, their notation, what we mean by zeros of a polynomial, and more. Let’s dive in.

What Is a Polynomial?

Here’s an example of a polynomial (with four terms).

9x^4 - 2x^3 - x + 1

A polynomial is an algebraic expression consisting of one or more terms connected to one another through operations of addition or subtraction. And each term can have a constant attached (multiplied) to a variable (or variables) with a non-negative exponent on it.

I repeat, in a polynomial, exponents of the variables can only be non-negative integers.

That means variables cannot appear as denominators (it would imply negative exponents) in a polynomial. Similarly, variable cannot be radicands (with fractional exponents) either.

So the following expressions are not polynomials.

$2x^5 + 3 {\color{Red} \sqrt{y}} + 8$
$x^2 + \frac{4}{ {\color{Red} x} } + 1$

And remember, this condition of non-negative exponents applies only to variables. Constants can be surds and they can be in the denominators. For example, here are two valid polynomials.

$x^3 + {\color{Teal} \sqrt{5}} y + 2xy - 1$
$y^2 + {\color{Teal} \frac{1}{3}} x + 9$

Alright, let’s see if you can correctly identify polynomials.

Example

Which of the following are polynomials?

\begin{align*} &(a) \hspace{1em} y^5 + 3y^2 - \sqrt{6}y^3 + 12 \\[1em] &(b) \hspace{1em} x^2 + \frac{2}{x^3} + 3 \\[1em] &(c) \hspace{1em} \sqrt{5}x^2yz + \frac{1}{2}yz + xy \\[1em] &(d) \hspace{1em} x \sqrt{y} - \sqrt{5}y + 8 \end{align*}

Solution

Expressions $\hspace{0.2em} (a) \hspace{0.2em}$ and $\hspace{0.2em} (c) \hspace{0.2em}$ are polynomials. They have constants and variables connected together by the math operations of addition, subtraction, and multiplication. And the powers of variables are non-negative integers. So all good.

The expression $\hspace{0.2em} (b) \hspace{0.2em}$ is not a polynomial because the second term has the variable $\hspace{0.2em} (x) \hspace{0.2em}$ in the denominator.

The expression $\hspace{0.2em} (d) \hspace{0.2em}$ is also not a polynomial because it has a variable under the radical sign ( $\hspace{0.2em} y \hspace{0.2em}$ in the first term).

Polynomials — Notation

The simplest way to denote a polynomial would be to assign a letter from the English alphabet (usually capital $\hspace{0.2em} P \hspace{0.2em}$ ) to it.

Here are a couple of examples.

\begin{align*} &P \, = \, x^2 + 1 \\[1em] &Q = x^3y^3 + 3xy^2 - y + 5 \end{align*}

However, you’ll also commonly see something like this —

\begin{align*} &P(x) \, = \, x^2 + 1 \\[1em] &Q(x, y) = x^3y^3 + 3xy^2 - y + 5 \end{align*}

Here, the part within the parentheses tells us about the variable(s) present in the polynomial. So the first polynomial is a polynomial in $\hspace{0.2em} x \hspace{0.2em}$ , while the second one is a polynomial in $\hspace{0.2em} x \hspace{0.2em}$ and $\hspace{0.2em} y \hspace{0.2em}$ .

Value of a Polynomial

Say, we have a polynomial,

P(x) \, = \, 2x^2 - x + 3

So, in that context, what does $\hspace{0.2em} P(4) \hspace{0.2em}$ mean? It refers to the value of the polynomial is we substitute $\hspace{0.2em} 4 \hspace{0.2em}$ in place of the variable, $\hspace{0.2em} x \hspace{0.2em}$ .

That means —

\begin{align*} P( {\color{Red} 4} ) \, &= \, 2 \cdot {\color{Red} 4} ^2 - {\color{Red} 4} + 3 \\[1em] &= \, 31 \end{align*}

Example

Find $\hspace{0.2em} P(-3) \hspace{0.2em}$ and $\hspace{0.2em} P(0)$ , if $\hspace{0.2em} P = 5x^2 - x + 2 \hspace{0.2em}$ .

Solution

To get the value of $\hspace{0.2em} P(-3) \hspace{0.2em}$ , we need to put $\hspace{0.2em} -3 \hspace{0.2em}$ in place of $\hspace{0.2em} x \hspace{0.2em}$ in $\hspace{0.2em} P(x) \hspace{0.2em}$ . So,

\begin{align*} P( {\color{Red} -3} ) \, \, &= \, \, 5 \cdot {\color{Red} (-3)} ^2 - {\color{Red} (-3)} + 2 \\[1em] &= \, \, 50 \end{align*}

Similarly,

\begin{align*} P( {\color{Red} 0} ) \, \, &= \, \, 5 \cdot {\color{Red} 0} ^2 - {\color{Red} 0} + 2 \\[1em] &= \, \, 2 \end{align*}

Degree of a Polynomial

When it comes to the degree of a polynomial, we need to start with what is meant by “the degree of a term”.

The degree of a term is the sum of the exponents of the variables in it.

For example, the degree of the term below is $\hspace{0.2em} 9 \hspace{0.2em}$ .

Remember, if a variable doesn’t carry a visible exponent, its exponent is $\hspace{0.2em} 1 \hspace{0.2em}$ . So, the exponent on $\hspace{0.2em} y \hspace{0.2em}$ is $\hspace{0.2em} 1 \hspace{0.2em}$ (above).

Now, the degree of a polynomial is the highest of the degrees of its terms.

Consider the following polynomial.

Below each term, you can see its degree. The highest of the four degrees is $\hspace{0.2em} 5 \hspace{0.2em}$ . So that’s the degree of the polynomial.

Here’s another example.

x^7 + x^3 - x^2 + 12

This polynomial has only one variable, $\hspace{0.2em} x \hspace{0.2em}$ . So the degree of each term is simply equal to the exponent of $\hspace{0.2em} x \hspace{0.2em}$ in the term. As such, the degree of the polynomial would be $\hspace{0.2em} 7 \hspace{0.2em}$ .

One takeaway from this is — if a polynomial has only one variable, its degree is equal to the largest exponent of the variable in it.

Standard Form of a Polynomial

The standard form of a polynomial is one in which its terms are arranged in decreasing order of their degrees.

Here’s a polynomial that’s not in its standard form.

Let’s rearrange the terms so we get the polynomial in its standard form.

See how the terms with higher degrees appear before others? Here’s another example of a polynomial in its standard form.

2y^3 + 3y^2 + y + 1

Leading Term and Leading Coefficient

The term with the highest degree in a polynomial is known as its leading term. And its coefficient is known as the leading coefficient.

When a polynomial is written in its standard form, the first term would be the leading term.

Example

Find the degree of the polynomial below and write it in its standard form. Also, what is its leading coefficient?

3y^2 - 5y - 2y^4 - 6y^5 + 1

Solution

Here we have a polynomial with only one variable, $\hspace{0.2em} y \hspace{0.2em}$ . So, the exponent of $\hspace{0.2em} y \hspace{0.2em}$ in each term gives us the term's degree.

Now, the term with the highest degree is the fourth one, with a degree of $\hspace{0.2em} 5 \hspace{0.2em}$ . So the degree of the polynomial is $\hspace{0.2em} 5 \hspace{0.2em}$ .

To get the polynomial in its standard form, let's arrange the terms in decreasing order of their degrees.

-6y^5 - 2y^4 + 3y^2 - 5y + 1

And finally, we know the leading coefficient is that of the term with the highest degree. So here, it's $\hspace{0.2em} -6 \hspace{0.2em}$ .

Some Special Polynomials

Certain kinds of polynomials present themselves so frequently, it makes sense to give them special names or labels.

Based on the Number of Terms

Depending on the number of terms, polynomials are often classified as monomials (a single term), binomials (two terms), trinomials (three terms), quadrinomials (four terms) and so on.

Type	# Terms	Example
Monomial	$\hspace{0.2em} 1 \hspace{0.2em}$	$\hspace{0.2em} -2a \hspace{0.2em}$
Binomial	$\hspace{0.2em} 2 \hspace{0.2em}$	$\hspace{0.2em} y + 5 \hspace{0.2em}$
Trinomial	$\hspace{0.2em} 3 \hspace{0.2em}$	$\hspace{0.2em} x^2 + 4x - 2 \hspace{0.2em}$

Based on the Degree

A polynomial can also be categorized on the basis of its degree. Here are some of the most common types.

Type	Degree	Example
Constant Polynomial	$\hspace{0.2em} 0 \hspace{0.2em}$	$\hspace{0.2em} 5 \hspace{0.2em}$
Linear Polynomial	$\hspace{0.2em} 1 \hspace{0.2em}$	$\hspace{0.2em} 3x - 1 \hspace{0.2em}$
Quadratic Polynomial	$\hspace{0.2em} 2 \hspace{0.2em}$	$\hspace{0.2em} x^2 + 4x - 2 \hspace{0.2em}$
Cubic Polynomial	$\hspace{0.2em} 3 \hspace{0.2em}$	$\hspace{0.2em} x^3 + 7x \hspace{0.2em}$
Quartic Polynomial	$\hspace{0.2em} 4 \hspace{0.2em}$	$\hspace{0.2em} x^4 + 5x^2 - 1 \hspace{0.2em}$
Quintic Polynomial	$\hspace{0.2em} 5 \hspace{0.2em}$	$\hspace{0.2em} x^5 \hspace{0.2em}$

Observe how it’s only the degree of the polynomial that matter. Not the number of terms or anything else.

Alright, let's look at a few of these in a little more detail.

Constant Polynomials

A zeroth degree polynomial is also known as a constant polynomial.

P = 8

Zeroth degree means there are no indeterminates/variables to affect the value of the polynomial. So its value remains constant. Hence, the name.

Linear Polynomials

Linear polynomials are polynomials of the first degree. For example —

Q = 2y + 9

They are called linear because when plotted on a graph, functions of linear polynomials produce a straight line.

Quadratic Polynomials

A polynomial of the second degree is known as a quadratic polynomial.

\begin{align*} &\text{R}_1 \, = \, x^2 - 5x - 1 \\[1.3em] &\text{R}_2 \, = \, 3x^2 + 16 \end{align*}

You’ll be seeing linear and quadratic polynomials a lot as you start working with equations.

Zeros of a Polynomial

The value(s) of $\hspace{0.2em} x \hspace{0.2em}$ for which the polynomial $\hspace{0.2em} P(x) \hspace{0.2em}$ becomes $\hspace{0.2em} 0 \hspace{0.2em}$ , are known as zeros of the polynomial.

For example, consider this polynomial.

P(x) \, = \, x^2 - 7x + 10

Let’s see what happens if we put $\hspace{0.2em} 2 \hspace{0.2em}$ or $\hspace{0.2em} 5 \hspace{0.2em}$ in place of $\hspace{0.2em} x \hspace{0.2em}$ .

\begin{align*} P( {\color{Red} 2} ) \, &= \, {\color{Red} 2} ^2 - 7 \cdot {\color{Red} 2} + 10 \\[1em] &= \, 0 \end{align*}

\begin{align*} P( {\color{Red} 5} ) \, &= \, {\color{Red} 5} ^2 - 7 \cdot {\color{Red} 5} + 10 \\[1em] &= \, 0 \end{align*}

In both cases, the value of the polynomial becomes $\hspace{0.2em} 0 \hspace{0.2em}$ . So, $\hspace{0.2em} 2 \hspace{0.2em}$ and $\hspace{0.2em} 5 \hspace{0.2em}$ are zeros of the given polynomial, $\hspace{0.2em} P(x) \hspace{0.2em}$ .

Like/Unlike Terms

We have already talked about how each term in a polynomial can be seen as having two parts — the coefficient and the variables (along with their exponents).

Like terms in a polynomial are those that carry the same variables (with the same exponents on them).

Terms that are not like are known as “unlike terms”.

Here are a few example of like and unlike terms.

The Importance

The idea of like and unlike terms becomes important when adding or subtracting polynomials. We can combine two or more like terms into one by adding/subtracting them together. For example —

\begin{align*} {\color{Red} 3} x^2y + {\color{Red} 5} x^2y \, &= \, ( {\color{Red} 3} + {\color{Red} 5} )x^2y \\[1em] &= \, {\color{Red} 8} x^2y \end{align*}

But we can’t do this with unlike terms. So the following binomial cannot be simplified further.

3x^2y + 5xy

Note — It doesn’t matter whether the terms are like or unlike when multiplying or dividing them.

And with that, we come to the end of this tutorial on polynomials. Until next time.