Two questions that sound almost silly until you try to answer them cleanly. Is the single number 7 a polynomial? And is the single number 0 a polynomial? The quick answer is yes to both, but 0 has a subtle wrinkle the definition does not fully cover — its degree sits in a strange place, and different textbooks handle it differently. This article works through the two cases, one at a time, and settles the wrinkle.
The definition, one more time
Read the definition slowly. A polynomial in x is a finite sum of terms, each of the form a_k \cdot x^k, where
- k is a non-negative integer (0, 1, 2, 3, \dots), and
- a_k is a coefficient (any real number).
Nothing in this sentence insists that k be positive or that the number of terms be more than one. A polynomial with exactly one term is allowed — it is just a monomial. A polynomial with k = 0 is allowed — that is the constant case, since x^0 = 1 for any nonzero x, and the convention extends this to x = 0 as well. So a single constant c fits the definition perfectly: write it as c \cdot x^0 and you have a one-term sum with k = 0 and a_0 = c. Valid polynomial.
So 7 is a polynomial of degree 0
Take p(x) = 7. Rewrite it as p(x) = 7 \cdot x^0. There is exactly one term. The coefficient is 7. The exponent of the variable is 0. The degree of the polynomial — the highest exponent of x with a nonzero coefficient — is 0. The polynomial is well-defined for every real x, and its value is always 7. Its graph is a horizontal line sitting seven units above the x-axis.
So 7 is not only a polynomial; it is a specific, named kind of polynomial: a constant polynomial, degree 0. The same holds for -3, for \pi, for \sqrt{2}, for any single real number that is not zero.
What is 0, then?
Now take p(x) = 0. Zero as a polynomial. Is that a thing?
Yes, and it has a name: the zero polynomial. It is the polynomial that evaluates to 0 for every input — p(3) = 0, p(-17) = 0, p(\pi) = 0, p(10^{100}) = 0. You can think of it in two equivalent ways:
- The sum of zero terms (an empty sum, which by convention equals 0).
- A sum where every coefficient is zero: 0 = 0 \cdot x^0 + 0 \cdot x^1 + 0 \cdot x^2 + \dots (truncated to finitely many terms, all with coefficient 0).
Both descriptions give the same object. The zero polynomial is the additive identity of the set of polynomials — the element you can add to any polynomial without changing it. Every algebraic system needs such an element, and for polynomials this is it. So yes, 0 is a polynomial. The definition does not reject it.
Why 0's degree is weird
Here is the wrinkle. The degree of a polynomial is defined as the highest power of x with a nonzero coefficient. For 3x^2 + 5x - 7, the highest-power-with-nonzero-coefficient is 2. Degree = 2. Easy.
For the zero polynomial, though, there is no nonzero coefficient anywhere. Every coefficient is zero. There is no "highest power with nonzero coefficient" to speak of, because there is no such power at all. The definition of degree simply does not produce a number for the zero polynomial.
Different textbooks resolve this in different ways.
- Undefined. The most common convention, especially in school textbooks. "The zero polynomial has no degree." Simplest to state; just flag 0 as a special case and move on.
- -\infty. Common in abstract algebra and higher mathematics. Assigning \deg(0) = -\infty makes certain theorems universally true (more on this in a moment).
- -1. A third convention, used occasionally in computer science and some coding-theory texts. Pragmatic but less principled.
Your Class 9 or Class 10 textbook almost certainly uses "undefined." Your college abstract algebra textbook will probably use -\infty. Both are correct — they are different conventions for the same underlying object.
Why the -\infty convention is popular in math
The motivation behind \deg(0) = -\infty is to keep one clean rule true without exception.
The rule is: \deg(p \cdot q) = \deg(p) + \deg(q). Multiply two polynomials, the degree of the product equals the sum of the degrees. For any two nonzero polynomials this works automatically — the leading terms multiply, producing the new leading term. Fine.
But now let q = 0. Then p \cdot q = p \cdot 0 = 0, which is the zero polynomial. For the rule to still hold, you need \deg(0) = \deg(p) + \deg(0), which means \deg(0) should be something that absorbs anything you add to it. The only value with that property, inside the extended real numbers, is -\infty:
So set \deg(0) = -\infty, and the product rule becomes universal. This is the whole reason the convention exists. It is a bookkeeping choice that keeps one important identity clean.
Nonzero constants: degree 0
To be completely clear: every nonzero constant is a polynomial of degree 0, no wrinkle.
- 7 has degree 0. Coefficient of x^0 is 7, all other coefficients are zero.
- -3/5 has degree 0. The coefficient being a fraction is fine — only the exponent has to be an integer.
- \pi has degree 0. Irrational coefficients are allowed.
- \sqrt{2}, e, -17.4 — all degree 0.
- i (the imaginary unit, if you are working with complex-coefficient polynomials) also has degree 0.
The pattern: any single nonzero number, written by itself, is a degree-0 polynomial. The graph is a horizontal line that is not the x-axis.
The zero polynomial and constant polynomials are different
It is tempting to say "constants are degree 0, so 0 should be degree 0 too." That is the trap. The constant polynomials of degree 0 and the zero polynomial are different animals:
- Constant polynomial: any a_0 \neq 0. Degree 0. Graph is a horizontal line above or below the x-axis.
- Zero polynomial: a_0 = 0 (and every other coefficient is also 0). Degree undefined (or -\infty). Graph is the x-axis itself.
The line y = 7 is not the x-axis. The line y = 0 is the x-axis. That geometric difference maps onto the algebraic difference. Theorems about polynomials typically treat these two cases separately, which is why the zero polynomial is often carved out as an exception.
Why this matters — theorems that exclude the zero polynomial
Here is one place the distinction bites. The standard theorem says: a polynomial of degree n has at most n roots. Take this literally with the zero polynomial. The zero polynomial evaluates to 0 at every real number. Every x is a root. That is infinitely many roots — more than any finite degree could possibly allow.
The way textbooks handle this is to restate the theorem with an explicit exclusion: a nonzero polynomial of degree n has at most n distinct roots. The zero polynomial is the one exception, and the theorem sidesteps it by definition. This is a recurring pattern — theorems about polynomials almost always say "nonzero polynomial" somewhere in their hypothesis, and that phrase exists precisely to rule out the weird case of 0.
Another subtlety — leading coefficient of the zero polynomial
While you are here, another object that misbehaves on the zero polynomial is the leading coefficient — the coefficient of the highest-degree term. For any nonzero polynomial, this is unambiguous: strip off the leading term, read its coefficient, done. For the zero polynomial, though, there is no leading term — no nonzero coefficient at all. So "leading coefficient of 0" is, like its degree, undefined. Any result that relies on the leading coefficient (end behaviour, monic-ness, division algorithms) also has to say "assume p is nonzero" before it can even get started.
Quick drill
Decide whether each of the following is a polynomial, and if so, name its degree.
- 7 → yes, degree 0.
- 0 → yes, degree undefined (or -\infty, depending on your textbook).
- -3 → yes, degree 0.
- 2x^3 → yes, degree 3.
- x^0 → yes, degree 0 (it equals 1).
- 0 \cdot x^5 → yes, but this is the zero polynomial (every coefficient is 0, even though it is written with an x^5 on the page). Degree undefined.
The last one is a favourite trick in exam questions. Just because you write 0 \cdot x^5 does not give you a degree-5 polynomial; the coefficient of x^5 is 0, and you have to look for the highest nonzero coefficient. There is none. Zero polynomial.
Summary
- Nonzero constants like 7, -3, \pi, \sqrt{2}: polynomials of degree 0. Plain and well-behaved.
- Zero (the number, equivalently the zero polynomial): also a polynomial, but with a special degree that is either undefined or assigned the value -\infty, depending on which convention your textbook follows.
- Neither breaks the polynomial definition. The zero polynomial is just the one case where the usual degree rule cannot be applied verbatim and a convention has to step in.
So: 7 is a polynomial of degree 0. 0 is a polynomial whose degree is "special" — know your textbook's convention. Both are in.