What Is (and Isn’t) a Polynomial
In other words, a monomial is a variable to some positive whole number power with (perhaps) a real number out in front. Doesn’t matter what form that number takes; could be a decimal, could be a fraction, etc.
Ask yourself whether each expression in the leftmost column is a monomial based on the definition above.
| Expression | Monomial? | Why/Why Not |
| \(3x^4\) | Yes | The power on \(x\) is a natural number and the coefficient \(3\) is a real number. |
| \(5x^{\frac{1}{2}}\) | No | The power is not a natural number |
| \(x^{10}\) | Yes | the coefficient out in front of the variable is implicitly \(1\) (i.e. \(x^{10}=1\cdot x^{10}\)) and so is a real number out front. Also, \(10\) is a positive whole number. |
| \(x\) | Yes | \(x\) can be written as \(1\cdot x^1\), giving us the required form |
| \(30\) or \(0\) | Yep | \(30\) or \(0\) can be written as \(30x^0\) and \(0x^1\), respectively. |
| \(39x^{4.1}\) | Nope | While the coefficient \(-39\) is a real number, the exponent \(4.1\) is not a natural number or 0. |
| \(4x^{-6}\) | Nope | The exponent \(-6\) is not a natural number or \(0\). |
| \(\frac{1}{x^{-7}}\) | Yes! | Note that \(\frac{1}{x^{-7}}\) can be rewritten as \(x^7\) using exponent rules. Thus \(\frac{1}{x^{-7}}\) is a monomial |
| \(\frac{5}{x^6}\) | Nope! | Even by fraction rules, you’d have \(\frac{5}{x^6}=5x^{-6}\), and the exponent is not a natural number. |
| \(-123.523x^{1278329.0}\) | Yep | The coefficient \(-123.523\) is a real number and the exponent \(1278329.0\), is a natural number even though it’s written as a decimal. |
Note that the prefix mono means “one” (indicating only one term in a monomial), and the prefix poly means “many” (indicating several terms in the polynomial).
Ask yourself whether each expression in the leftmost column is a polynomial based on the definition above.
| Expression | Monomial? | Why/Why Not |
| \(5x^{\frac{1}{2}}\) | No | This isn’t even a monomial because the exponent is a fraction, so it cannot be a polynomial either. |
| \(3x^4-3x^3+7x-1\) | Yes | Note that each term \(3x^4\), \(3x^3\), \(7x\) and \(1\) is a monomial, and we are adding and subtracting these terms. therefore we have a polynomial. |
| \(3.15x^4-\frac{1}{3}x^3+10\) | Yes | All exponents present in this expression are natural numbers (or \(0\) in the case of the last term \(10\). The coefficients are all real numbers, so we have a sum of monomials |
| \(x^{12}\) | Yep | Technically, one can write \(x^{12}\) as \(1\cdot x^{12}+0\), so that it becomes clear that we have a sum of monomials. |
| \(39x^{4.1}\) | Nope | \(x^7\), \(3x^3\) and \(50x\) are all monomials, but the last term \(10x^{3.01}\) is not a monomial (because the power is not a natural number or zero). |
| \(x+\frac{1}{x^{2}}\) | No | The first term \(x\) is indeed a monomial, but \(\frac{1}{x^2}\) is NOT a monomial. |
| \(5\) | Yes! | In general, any real number you choose, say \(r\), can be written as \(r\cdot x^1+0\) which is a sum of monomials. |
| \(x^2-x+\frac{4}{x^{-6}}\) | Yes! | the first two terms are certainly monomials. By fraction rules, your last term can be written as \(4x^6\) which also is a monomial. |
| \(41.2x^3-8x^2+2x+\frac{5}{3}\) | Yep | All terms are clearly monomials |
IMPORTANT NOTE: All monomials are also polynomials (just add zero to any given monomial to see that you now have a sum of monomials; try it yourself!).
For instance, the degree of the polynomial \(1.23x^4-2x^3+1\) is \(4\) because that’s the highest power one sees in the expression.
Note that the term “degree” only applies to polynomials. One cannot find the degree of an expression like \(3x^{3.15}-6x^2+2x\) because the expression is not a polynomial.