Zero Polynomial — Definition, Degree, and Examples

What Is the Zero Polynomial?

The zero polynomial is the polynomial in which every coefficient equals $0$. It collapses to the single value $0$, so we write it as $P(x) = 0$, and it returns $0$ no matter what number you substitute for the variable.

It is a constant polynomial — a polynomial with no variable terms — but it is the one constant polynomial that behaves differently from all the others. A nonzero constant like $P(x) = 7$ has degree $0$. The zero polynomial does not, and that single exception is the whole story.

Zero polynomial in symbols: $P(x) = 0$, or equivalently $0 \cdot x^n + 0 \cdot x^{n-1} + \dots + 0$ for any $n$.

Examples of the Zero Polynomial

The examples below build from spotting the zero polynomial to reasoning about its degree and its zeros.

Example 1

Is $P(x) = 0x^2 + 0x + 0$ the zero polynomial?

Every coefficient is $0$. The expression collapses to $0$. So $P(x) = 0$ for every $x$.

Yes — this is the zero polynomial, dressed up to look like a quadratic.

Example 2

A student is asked for the degree of $P(x) = 0$. They answer "0, because it is a constant." Is that right?

Here is the tempting path first.

A nonzero constant such as $5$ is written $5x^0$, so its degree is $0$. The reasoning says: $0$ is a constant, so its degree must also be $0$.

That looks airtight — but watch it break. The degree is defined as the exponent of the highest power that has a nonzero coefficient. The zero polynomial has no nonzero coefficient at all, so there is no "highest power" to point to.

The correct answer: the degree of the zero polynomial is undefined.

Example 3

Find the zeros of the zero polynomial.

A zero of a polynomial is any value of $x$ for which $P(x) = 0$. For the zero polynomial, $P(x) = 0$ is true for every $x$.

So every real number is a zero. The zero polynomial has infinitely many zeros — another way it stands apart from ordinary polynomials.

Example 4

Is $Q(x) = x^2 - x^2$ the zero polynomial?

Combine like terms first. $x^2 - x^2 = 0$.

Yes. After simplifying, $Q(x) = 0$, so it is the zero polynomial even though it did not start out looking like one. Always simplify before judging.

Example 5

Add the zero polynomial to $R(x) = 3x + 4$.

$R(x) + 0 = 3x + 4 + 0$ $= 3x + 4$

Nothing changes. The zero polynomial is the additive identity — adding it leaves any polynomial exactly as it was, the same role $0$ plays for ordinary numbers.

Example 6

Multiply the zero polynomial by $S(x) = 2x^3 - 7$.

$0 \cdot (2x^3 - 7) = 0$

Every product term carries a factor of $0$, so the whole product is $0$. Multiplying any polynomial by the zero polynomial gives the zero polynomial back.

Why the Degree Is Left Undefined

The degree of the zero polynomial is not a fact waiting to be measured; it is a choice mathematicians made to keep the rules consistent.

The degree of a polynomial is the largest exponent whose coefficient is nonzero. Run that definition on $0$ and it stalls — there is no nonzero coefficient anywhere. So which power should we call the "highest"?

• $0$ could be read as $0 \cdot x^0$ (suggesting degree $0$).

• It could equally be read as $0 \cdot x^5$ (suggesting degree $5$).

• Or $0 \cdot x^{100}$, or any power at all.

Because the value $0$ gives no way to prefer one over another, the honest answer is that the degree simply isn't defined. Some textbooks assign it $-\infty$ or $-1$ instead — and there is a reason for that convention.

The degree rule for products says $\deg(P \cdot Q) = \deg P + \deg Q$. If we want that rule to keep working when one factor is the zero polynomial (whose product with anything is $0$), the degree has to behave like $-\infty$, because $-\infty + n = -\infty$ for any $n$. Wolfram MathWorld and the Wikipedia entry on the degree of a polynomial both note that $-\infty$ is the choice that keeps the algebra clean. For school-level work, undefined is the standard answer; $-\infty$ is the reason the convention exists.

Zero Polynomial vs Zero of a Polynomial

These two phrases share the word "zero" but name completely different objects, and mixing them up is the single biggest source of wrong answers on this topic.

• The zero polynomial is a whole function: $P(x) = 0$, the polynomial whose every coefficient is $0$.

• A zero of a polynomial is a single number: a value of $x$ that makes some polynomial equal $0$ — what you usually call a root.

So $x - 3$ is not a zero polynomial; it merely has a zero at $x = 3$. The zero polynomial, by contrast, is zero everywhere, which is exactly why it has infinitely many zeros of its own.

Properties of the Zero Polynomial

The zero polynomial obeys a short, memorable list of rules — most of them mirroring the number $0$ itself.

• All coefficients are $0$. It equals $0$ for every input, so its graph is the x-axis.

• Degree is undefined. There is no nonzero coefficient to anchor a highest power (some texts use $-\infty$).

• Additive identity. $P(x) + 0 = P(x)$ for every polynomial $P(x)$: adding it changes nothing.

• Absorbing under multiplication. $0 \cdot P(x) = 0$: multiplying any polynomial by it returns the zero polynomial.

• Infinitely many zeros. Every real number is a zero, since $P(x) = 0$ holds for all $x$.

• Constant, but exceptional. It is a constant polynomial, yet the only one whose degree is not $0$.

• No leading coefficient. With no highest-degree term, there is nothing to be the leading coefficient.

Where Zero Polynomials Go Wrong

Mistake 1: Calling the degree $0$

Where it slips in: Right after learning that a nonzero constant like $7$ has degree $0$. The word "constant" gets over-applied.

Don't do this: Write "degree of $0$ is $0$" because $0$ is a constant.

The correct way: The degree counts the highest power with a nonzero coefficient. The zero polynomial has none, so its degree is undefined. The instinct here is to treat $0$ as just another constant — but the degree rule keys off nonzero coefficients, and that is exactly the ingredient $0$ is missing.

Mistake 2: Confusing the zero polynomial with a zero of a polynomial

Where it slips in: The words sound almost identical. A zero of a polynomial is a root — an $x$ that makes $P(x) = 0$. The zero polynomial is the whole function $P(x) = 0$.

Don't do this: Say "the polynomial $x - 3$ is a zero polynomial because $x = 3$ is its zero."

The correct way: $x - 3$ has one zero ($x = 3$) but is not the zero polynomial. The rusher who reads only the shared word "zero" collapses two different ideas; the reader who checks the definition keeps them apart. This mix-up is the single most common source of wrong answers on this topic.

Mistake 3: Forgetting to simplify first

Where it slips in: An expression like $x^2 - x^2$ or $(x+1) - (x+1)$ looks like a real polynomial until you combine terms.

Don't do this: Report the degree as $2$ for $x^2 - x^2$ because you see an $x^2$.

The correct way: Simplify first, since $x^2 - x^2 = 0$, and only then read off the degree.

The Mathematicians Behind the Zero Polynomial

The convention that the zero polynomial has degree $-\infty$ traces to the algebraic formalism developed by Carl Friedrich Gauss (1777–1855, Germany), whose work on polynomial rings made the degree-of-a-product rule central. The full ring-theoretic framing that makes $-\infty$ natural was later formalised by Emmy Noether (1882–1935, Germany), a founder of modern abstract algebra. See MacTutor for Gauss and Noether.

Conclusion

• The zero polynomial is $P(x) = 0$: every coefficient is $0$ and it equals $0$ for all inputs.

• Its degree is undefined (some texts use $-\infty$) because there is no nonzero coefficient to anchor a highest power.

• It has infinitely many zeros — every real number qualifies.

• It is the additive identity for polynomials and absorbs any product to $0$.

• Simplify an expression before judging whether it is the zero polynomial.

To take the zero polynomial further with a teacher, explore Bhanzu's algebra tutor, help with algebra, or browse math classes online. Want to build this with a live trainer? Book a free demo class.

Now check your understanding: classify $0x^3 + 0x$, $x^3 - x^3$, and $x - 5$ as zero polynomial or not, and state the degree of each. If the degree of $0$ trips you up, return to the "Why the Degree Is Left Undefined" section.

Read More

• Polynomials: definition, terms, and types

• Types of polynomials

• Degree of a polynomial

• Standard form of a polynomial

• Cubic polynomials