# Zero Polynomial — Definition, Degree, and Examples

TL;DR

The zero polynomial is the polynomial whose every coefficient is 0, so it equals 0 for every input and is written P(x)=0. Its degree is undefined, it has infinitely many zeros, and this article shows why through worked examples and the mistakes students make.

## 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⋅x^n + 0⋅x^{n−1} + ⋯ + 0 for any n.

## Examples of the Zero Polynomial

### 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⋅(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⋅x^0 (suggesting degree 0).
- It could equally be read as 0⋅x^5, or 0⋅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**.

## 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**.

|  | Zero polynomial | Zero of a polynomial |
| --- | --- | --- |
| What it is | An entire polynomial | A single input value (a root) |
| Symbol | P(x)=0 | x=c with P(c)=0 |
| Example | P(x)=0 | For P(x)=x−3, the zero is x=3 |
| How many | One such polynomial exists | A degree-n polynomial has at most n of them |

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 −∞).
- **Additive identity.** P(x)+0=P(x) for every polynomial P(x): adding it changes nothing.
- **Absorbing under multiplication.** 0⋅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 but is not the zero polynomial. 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 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.

## 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 −∞) 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.