What Cannot Be A Polynomial? (11 Key Things To Know)

Polynomials come up a lot in algebra, but it is still a good idea to know the definition.  That way, you know what a polynomial can contain and what it cannot.

So, what cannot be a polynomial?  An expression with a variable with negative or fractional exponents, division by a variable, or a variable inside a radical is not a polynomial.  However, a polynomial may contain coefficients that are negative, fractions, or even radicals, as long as the polynomial is defined over the real numbers.

Of course, it takes a little practice to be able to quickly recognize an expression that is not a polynomial.

In this article, we’ll answer some common questions about what a polynomial can and cannot be.  We’ll also look at some examples to make the concepts more clear.

Let’s get started.

What Cannot Be A Polynomial?

A polynomial cannot have any of the following:

• A variable with a negative exponent.
• Division by a variable (this can lead to negative exponents).
• A variable with a fractional exponent (unless the fraction reduces to a whole number).
• A variable inside a radical (this can lead to fractional exponents).

The sections below answer some more specific questions about polynomials and what they contain.  There are also some specific examples to help illustrate the ideas.

Can A Polynomial Have A Negative Exponent?

A polynomial cannot have a negative exponent.  By the definition of a polynomial, the exponent of a variable in any term of a polynomial must be a nonnegative integer, such as 0, 1, 2, 3, 4, … etc.

In other words, for any polynomial, the power of a variable in a term is either:

• A positive whole number (1, 2, 3, 4, 5, … etc.)
• Zero

There are no fractions, decimals, or negative numbers in the exponents of variables.

Example 1: A Polynomial

For example, consider the polynomial

• 2x4 – 5x3 + 8x2 + 9x + 7

Each exponent of x in a given term is a nonnegative integer: 4 for the first term, 3 for the second term, 2 for the third term, 1 for the fourth term, and 0 for the 5th term (since 7 is really 7x0).

Example 2: Not A Polynomial Due To A Negative Exponent

Consider the expression:

• 4x-1 + 2

This is not a polynomial, since x has a negative exponent (a value of -1) in the first term.

Can A Polynomial Have A Variable In The Denominator?

A polynomial cannot have a variable in the denominator of any term.  In other words, we are only adding, subtracting, and multiplying powers of x – we are not dividing them.

Example 1: Not A Polynomial Due To A Variable In The Denominator Of A Term

Consider the expression:

• 2 + 5/x

This is not a polynomial, since there is a variable in the denominator of a term.  Note that this expression is equivalent to having a negative exponent in the second term, since:

• 2 + 5/x = 2 + 5x-1

Example 2: Not A Polynomial Due To A Variable In The Denominator Of The Expression

Consider the expression:

• (5x + 3x2) / x2

This is not a polynomial, since we have a variable in the denominator of the expression.  Note that this expression is equivalent to one with a negative exponent as well, since:

• (5x + 3x2) / x2 = 5/x + 3 = 5x-1 + 3

Can A Polynomial Have A Square Root?

A polynomial cannot have a square root.  The reason is that this would involve a power that is not a whole number (since a square root is a power of 1/2).

Example 1: Not A Polynomial Due To A Square Root In One Term

Consider the expression:

• 2x + √x – 5

This is not a polynomial, since we have a square root in the second term.  Note that this expression is equivalent to one with a variable that has a fraction exponent, since:

• 2x + √x – 5 = 3x + x1/2 – 5

Example 2: Not A Polynomial Due To A Square Root In The Expression

Consider the expression:

• √(x – 8) + 4

This is not a polynomial, since we have a square root in the first term.  Note that this expression is equivalent to one with a variable that has a fraction exponent, since:

• √(x – 8) + 4 = (x – 8)1/2 + 4

Can A Polynomial Have A Fraction Exponent?

A polynomial cannot have a fraction exponent if the fraction does not reduce to a whole number.

Example 1: A Polynomial With A Fraction Exponent That Reduces To A Whole Number

Consider the expression:

• x6/3 – 9

We can reduce 6/3 to 2 and rewrite the expression as:

• x2 – 9

This is a polynomial, since the powers of x (2 in the first term, 0 in the second term) are nonnegative whole numbers.

Example 2: Not A Polynomial Due To A Fraction Exponent

Consider the expression:

• x1/3 – 8

This is not a polynomial, since we have a fraction exponent (a value of 1/3) in the first term.  Since 1/3 cannot be reduced to a whole number, the expression is not a polynomial.

Can A Polynomial Have A Radical?

A polynomial cannot have a radical, since this would mean that there are powers of a variable that are not whole numbers.

Example 1: Not A Polynomial Due To A Radical In A Term

Consider the expression:

• ∛x + 5

This is not a polynomial, since we have a radical in the first term.  Note that this expression is equivalent to one with a variable that has a fraction exponent, since:

• ∛x + 5 = x1/3 + 5

Example 2: Not A Polynomial Due To A Radical Over The Expression

Consider the expression:

• ∛(x3 – 8) + 1

This is not a polynomial, since we have a radical in the first term.  Note that this expression is equivalent to one with a variable that has a fraction exponent, since:

• ∛(x3 – 8) + 1 = (x3 – 8)1/3 + 1

Can A Polynomial Have A Fraction?

A polynomial can have a fraction in any of its coefficients, but not in any of the exponents of variables.

An exception would be a polynomial with integer coefficients.  In that case, the polynomial could not have any fractions at all – not in the coefficients and not in the exponents.

Example 1: A Polynomial With A Fraction Coefficient

Consider the expression:

• (1/3)x2 – 5x + 1

This is a polynomial, even though we have a fraction (1/3) in the coefficient of the first term.  The reason is that all of the exponents are whole numbers for the variable x (2 in the first term, 1 in the second term, and 0 in the third term).

Example 2: Not A Polynomial Due To A Fraction Exponent

Consider the expression:

• 5x2/5 – 8

This is not a polynomial, since the exponent in the first term is a fraction (2/5) that does not reduce to a whole number.

Can A Polynomial Be A Fraction?

A polynomial can be a fraction in certain cases, and it can contain one or more terms that are fractions.  We just need to avoid variables in the denominators of these fractions.

Example 1: A Fraction That Is A Polynomial

Consider the expression:

• 2/3

Technically, this fraction is a polynomial, since it is just a constant term (with an exponent of zero on x).

Example 2: A Polynomial That Is A Fraction

Consider the expression:

• (x2 + 5x + 3) / 2

This is a polynomial, since we can easily separate the terms in the numerator and rewrite it as:

• (1/2)x2 + (5/2)x + (3/2)

Although each coefficient is a fraction, the powers of x are all whole numbers: 2 for the first term, 1 for the second term, and zero for the third term.

Example 3: A Fraction That Is Not A Polynomial

Consider the expression:

• (10x2 + 6x + 8) / 2x

This is not a polynomial, since we can easily separate the terms in the numerator and rewrite it as:

• 5x + 3 + 4/x

Although each coefficient is a whole number, the exponent on the third term is negative, since:

• 4/x = 4x-1

Can A Polynomial Have Two Variables?

A polynomial can have two variables.  In fact, a polynomial can have three variables or more – as many as you like.

However, the same rules apply for each variable: the exponent of each variable in any term must be a nonnegative integer.

Example 1: A Polynomial With Two Variables

Consider the expression:

• 5x2 + 2xy + 6y2

This is a polynomial, since the exponents are nonnegative integers in every term.  For example, we can rewrite the expression as:

• 5x2y0 + 2xy + 6x0y2

The table below shows the values of the exponents of each variable in each term:

Example 2: A Polynomial With Three Variables

Consider the expression:

• x3 + y3 + z3

This is a polynomial, since the exponents are nonnegative integers (all have values of 3 or zero) in every term.

Example 3: Not A Polynomial Due To Division Of Variables

Consider the expression:

• x / y

This is not a polynomial, since we are dividing by the variable y.  Although we are not dividing by the polynomial x, this expression is equivalent to one where y has a negative exponent:

• x / y = xy-1

Can A Polynomial Be Negative?

A polynomial can have negative coefficients – in fact, it is possible that every coefficient could be negative.  However, the exponents of the variables cannot be negative.

Example 1: A Polynomial That Has All Negative Coefficients

Consider the expression:

• -5x – 4y – 3z

This is a polynomial, since the powers of each variable are nonnegative integers.  Note that all of the coefficients are negative: -5 in the first term, -4 in the second term, and -3 in the third term.

Example 2: Not A Polynomial Due To Negative Exponent

Consider the expression:

• -7x + 4y – 3z-2

This is not a polynomial, since the power of z is negative (-2).

Can A Polynomial Have One Term?

A polynomial can have one term.  This one term could have any power of x, as long as the exponent is a nonnegative integer.

Example 1: A Constant Polynomial With One Term

Consider the expression:

• 5

This is a polynomial, since the power of the variable x is zero (5 = 5x0).

Example 2: A Linear Polynomial With One Term

Consider the expression:

• 4x

This is a polynomial, since the power of the variable x is 1 (4x = 4x1).

Example 3: A Quadratic Polynomial With One Term

Consider the expression:

• 7x2

This is a polynomial, since the power of the variable x is 2.

Can A Polynomial Have No Constant Term?

A polynomial can have no constant term.  Another way to say this is that the constant term is zero.

Example: A Polynomial With No Constant Term

Consider the expression:

• x2 + 2x

This is a polynomial with no constant term, since it can be rewritten as

• x2 + 2x + 0

Conclusion

Now you know more about what can and cannot be a polynomial.  You also know the answers to some common questions about polynomials and what they can contain.

You might be interested in learning about synthetic division (a shortcut for polynomial division) in my article here.