Zero Polynomial — Definition, Degree, and Examples
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.