Degree of a Polynomial: Definition, How to Find, Examples

What Is the Degree of a Polynomial?

The degree of a polynomial is the highest of the degrees of its individual terms, counting only terms with non-zero coefficients. A polynomial is a sum of terms, each one a number (the coefficient) multiplied by a variable raised to a whole-number power.

For a single-variable polynomial, the degree is just the largest exponent that appears. The degree of a single term, called its monomial degree, is the exponent on its variable. So in $5x^3 + 2x - 9$, the term degrees are $3$, $1$, and $0$, and the polynomial's degree is $3$.

Two terms quietly carry hidden exponents. A lone variable like $x$ means $x^1$, so its degree is $1$. A plain number like $7$ is $7x^0$, since $x^0 = 1$, so its degree is $0$. Spotting those two is half the skill.

Variable Glossary:

How Do You Find the Degree of a Polynomial?

You find the degree in three moves: tidy the polynomial, read the exponents, and take the largest. The first move matters more than it looks, because an untidied polynomial hides its real degree.

1. Combine like terms so each power appears only once.

2. Drop any term whose coefficient became zero.

3. Take the highest remaining exponent. That is the degree.

Take $5x^5 + 7x^3 + 2x^5 + 3x^2 + 8x + 4$. The two $x^5$ terms combine into $7x^5$, so the highest surviving power is $5$. The degree is $5$, even though the polynomial first looked like it had a clutter of competing powers.

What About Polynomials With More Than One Variable?

For a multivariable polynomial, the degree of each term is the sum of the exponents on all its variables, and the polynomial's degree is the largest of those sums. In $x^3 + 6x^2y^4 + 3y^2 + 5$, the term $6x^2y^4$ has degree $2 + 4 = 6$, which beats every other term. The degree is $6$.

This trips up students who scan for the single biggest exponent and stop. The biggest visible exponent here is $4$, but the answer is $6$. You have to add within a term before comparing across terms.

Names of Polynomials by Degree

Polynomials get names from their degree, and those names appear constantly in later algebra. A quadratic equation is just a degree-2 polynomial set equal to zero; a cubic polynomial is degree 3.

A useful rule of thumb sits inside this table: a polynomial of degree $n$ has at most $n$ real solutions and at most $n - 1$ turning points on its graph. That single fact is why the degree gets checked first.

The Two Special Cases: Constant and Zero Polynomials

Two polynomials break the simple "read the exponent" rule, and exam questions love them.

A constant polynomial is a non-zero number on its own, like $40$. It can be written $40x^0$, and since $x^0 = 1$, its degree is $0$. Every non-zero constant has degree zero.

The zero polynomial is the number $0$ by itself, and its degree is undefined (some textbooks write $-\infty$ to keep arithmetic rules consistent). It is not degree $0$. The constant $0$ has no non-zero term to take an exponent from, so there is nothing to read. Write "degree undefined" and move on.

Examples of Degree of a Polynomial

These six examples move from a one-glance read to multivariable terms and the zero-polynomial trap. The problem statement is bold; the working is not.

Example 1

Find the degree of $4x^2 + 3x + 9$.

The exponents are $2$, $1$, and $0$.

The highest is $2$.

Degree $= 2$ (quadratic).

Example 2

Find the degree of $6x^4 + 2x^3 + 3$, but first watch a common wrong move.

Wrong attempt: a student sees the coefficients $6$, $2$, $3$ and the powers $4$, $3$, $0$, then picks $6$ because it is the biggest number on the page.

That gives "degree $6$", which cannot be right: the largest exponent anywhere is only $4$, so the curve can turn at most three times. A degree of $6$ would need an $x^6$ term, and there isn't one.

Correct method: ignore coefficients entirely. The degree comes from exponents only.

The exponents are $4$, $3$, $0$.

Degree $= 4$.

Example 3

Find the degree of $5x^5 + 7x^3 + 2x^5 + 3x^2 + 8x + 4$.

Combine like terms: $5x^5 + 2x^5 = 7x^5$.

The tidied polynomial is $7x^5 + 7x^3 + 3x^2 + 8x + 4$.

The highest exponent is $5$.

Degree $= 5$.

Example 4

Find the degree of the multivariable polynomial $x^3 + 6x^2y^4 + 3y^2 + 5$.

Add exponents within each term: $x^3$ gives $3$; $6x^2y^4$ gives $2 + 4 = 6$; $3y^2$ gives $2$; the constant gives $0$.

The largest sum is $6$.

Degree $= 6$.

Example 5

Find the degree of $9$ and of $0$.

The number $9$ is a non-zero constant, $9 = 9x^0$.

Degree of $9 = 0$.

The number $0$ is the zero polynomial, with no non-zero term.

Degree of $0$ is undefined.

Example 6

A polynomial simplifies to $2x^4 - 2x^4 + 7x$. Find its degree.

Combine like terms: $2x^4 - 2x^4 = 0$, so both $x^4$ terms vanish.

What remains is $7x$.

The highest surviving exponent is $1$.

Degree $= 1$ (linear, not quartic).

Where Degree Earns Its Keep

The degree is not a label you memorise for a quiz; it is the first thing a working mathematician checks, because it sets the rules for everything that follows.

• It caps the number of solutions. The Fundamental Theorem of Algebra, proved by Carl Friedrich Gauss in 1799, says a degree-$n$ polynomial has exactly $n$ roots counted with multiplicity. Knowing the degree tells you how many answers to hunt for before you start.

• It predicts the graph's shape. A degree-2 polynomial is always a parabola; a degree-3 always has that signature S-bend. Engineers reading a polynomial's standard form glance at the degree to know what curve they are dealing with.

• It controls behaviour at the extremes. As $x$ runs off to large positive or negative values, the highest-degree term dominates everything else. A cubic eventually outgrows any quadratic, which is why degree alone decides which curve wins far from the origin.

The reason the degree comes first is that it converts a messy expression into a single predictive number. Without it, you would graph blindly and solve without knowing when to stop.

Where Students Trip Up on Degree of a Polynomial

Three mistakes account for most lost marks on this topic, and each one comes from reading the polynomial too fast.

Mistake 1: Picking the largest coefficient instead of the largest exponent

Where it slips in: When a polynomial has a big number out front, like $9x^2 + 3x$, the eye lands on the $9$.

Don't do this: Call the degree $9$ because $9$ is the biggest number visible.

The correct way: The degree comes from exponents only. In $9x^2 + 3x$ the degree is $2$. The coefficient never affects the degree. The first instinct here is to read the number in front of the variable as if it were the power, and that swap is the single most common source of wrong answers on this topic.

Mistake 2: Calling the degree of the zero polynomial zero

Where it slips in: A simplification collapses every term to zero, leaving just $0$, and the answer feels like it should be a number.

Don't do this: Write "degree of $0$ is $0$" by analogy with the constant case.

The correct way: The degree of the zero polynomial is undefined. A non-zero constant like $5$ has degree $0$; the number $0$ alone has no degree at all. The memorizer who learned "constants are degree 0" applies the rule one step too far and stumbles here, because $0$ is not an ordinary constant.

Mistake 3: Forgetting to combine like terms first

Where it slips in: A long polynomial repeats a power, like two separate $x^5$ terms, or hides a cancellation.

Don't do this: Read the highest exponent straight off the unsimplified expression.

The correct way: Combine like terms before you judge the degree. In $2x^4 - 2x^4 + 7x$, the $x^4$ terms cancel and the real degree is $1$, not $4$. Skipping the tidy-up step is exactly where the degree goes wrong.

Key Takeaways

• The degree of a polynomial is the highest exponent on its variable after combining like terms and dropping zero coefficients.

• For multivariable terms, add the exponents within each term, then take the largest sum across terms.

• A non-zero constant has degree $0$; the zero polynomial $0$ has degree undefined, never $0$.

• The degree never depends on coefficients, only on exponents.

• Degree by name: 0 constant, 1 linear, 2 quadratic, 3 cubic, 4 quartic, and a degree-$n$ polynomial has at most $n$ roots.

Practice This to Solidify Your Understanding

Find the degree of each: (1) $7x^3 - 2x + 1$, (2) $4x^2y^3 + x^5$, (3) the polynomial that simplifies to $3x^2 - 3x^2 + 6$, (4) the number $0$. Work each one by combining like terms first, then reading the exponents, never the coefficients. If you get stuck on the multivariable one, return to the section on more than one variable above.

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