The degree of a polynomial is the highest degree of any of its terms. For one variable, that is the largest exponent: 4x3 − 2x5 + 7x has degree 5. For a term with several variables, add their exponents, so 3x2y4 has degree 6. A nonzero constant such as 9 has degree 0.
In Spanish
grado de un polinomio
Also called
polynomial degree, order of a polynomial, highest power
Find each term’s degree, then keep the largest. The order the terms are written in does not matter.
Formal definition and notation
The degree of a nonzero term ax1e1 ⋯ xkek is the sum e1 + ⋯ + ek of its exponents, and the degree of a nonzero polynomial is the greatest degree among its terms after like terms are combined. A nonzero constant has degree 0; the zero polynomial is usually given no degree (or degree −∞). For one variable, the degree tells the most real roots the polynomial can have and, together with the leading coefficient, its end behavior.
Symbol
Read it as
Meaning
deg(p)
degree of p
the degree of polynomial p; deg(x4 − 1) = 4
Key facts
The terms do not have to be in order: in 7 + x4 − 3x6 the degree is 6, from the last term.
A polynomial of degree n has at most n real roots, and its graph has at most n − 1 turning points.
Even degree: both ends of the graph point the same way. Odd degree: the ends point opposite ways.
Multiplying polynomials adds their degrees; adding polynomials gives at most the larger degree.
How to find the degree
How to find the degree
degree = the largest (sum of exponents in a term)
one variable
the biggest exponent of x: deg(5 − x + 2x3) = 3
several variables
add exponents within each term, then take the biggest: deg(x2y + xy3) = 4
Simplify first. x3 + 2x − x3 looks like degree 3, but it simplifies to 2x, which has degree 1. For a product, add the degrees: (x2 + 1)(x3 − x) has degree 2 + 3 = 5.
Degree of a Polynomial examples, step by step
Example 1: One variable
Warm-up
What is the degree of 6x4 − x2 + 3?
Term degrees: 6x4 has 4, −x2 has 2, and 3 has 0.
The largest is 4.
So it is a quartic polynomial.
Answer: The degree is 4.
Example 2: Two variables
Core
What is the degree of 5a2b3 − 2ab + 7a4?
5a2b3: add exponents, 2 + 3 = 5.
−2ab: 1 + 1 = 2. 7a4: 4.
The largest term degree is 5.
Answer: The degree is 5 (from 5a2b3), even though no single exponent is 5.
Example 3: Degree of a product
Test-style
What is the degree of (2x3 + 1)(x2 − 4x)?
The highest-degree terms are 2x3 and x2.
Their product, 2x5, is the highest-degree term of the product.
So the degree is 3 + 2 = 5. (Expanded: 2x5 − 8x4 + x2 − 4x.)
Answer: The degree is 5.
Common mistakes
✗ MistakeSaying 4x3 − 2x5 + 7x has degree 3 because the first term has exponent 3.
✓ InsteadCheck every term: the highest exponent is 5, so the degree is 5.
The degree is the highest exponent anywhere, not the exponent of the first term. Writing in standard form (descending powers) puts it first.
✗ MistakeGiving 3x2y4 degree 4, the larger exponent.
✓ InsteadAdd the exponents: 2 + 4 = 6.
The degree of a term counts all its variable factors together.
✗ MistakeCounting terms: “x³ + x + 1 has 3 terms, so its degree is 3.”
✓ InsteadIts degree is 3 because of x³, not because there are 3 terms. x⁵ + 1 has 2 terms and degree 5.
The degree is the exponent of the highest-degree term; the leading coefficient is the number in front of it. In −2x5 + 4x3, the degree is 5 and the leading coefficient is −2.
A binomial has 2 terms whatever its degree: x9 + 1 is a binomial of degree 9.
Degree of a Polynomialvs.Degree of an angle
Degrees also measure angles (90°). The degree of a polynomial is an exponent count and has no unit.
Quick check: 3 questions
Question 1 of 3
What is the degree of 7x2 − 3x5 + x?
Show answer and explanation
Answer: B. The largest exponent is 5, from −3x5. 2 is only the first term’s exponent, 3 is the number of terms, and 7 is a coefficient.
Question 2 of 3
What is the degree of the polynomial 12?
Show answer and explanation
Answer: C. A nonzero constant is 12x0, so its degree is 0. 12 is its value, not its degree, and only the zero polynomial is left without a degree.
Question 3 of 3
What is the degree of x2y3 + 4xy?
Show answer and explanation
Answer: D.x2y3 has degree 2 + 3 = 5 and 4xy has degree 2, so the degree is 5. 3 takes the largest single exponent, and 6 multiplies 2 and 3 instead of adding.
Degree of a Polynomial: frequently asked questions
How do you find the degree of a polynomial?
Simplify it, find the degree of each term (its exponent, or the sum of its exponents if it has several variables), and take the largest. For 2x3 − 5x6 + 1, the term degrees are 3, 6 and 0, so the degree is 6. Writing the polynomial in standard form puts the highest-degree term first.
What is the degree of a constant?
A nonzero constant, such as 7 or −3, has degree 0, because it equals 7x0 and has no variable factor. Its graph is a horizontal line. The zero polynomial, 0, is a special case: most textbooks say it has no degree, because 0 = 0xn for every n.
Why does the degree of a polynomial matter?
It tells you the shape of the graph and how many answers to expect. A degree-2 polynomial graphs as a parabola and has at most 2 real roots; a degree-3 polynomial has at most 3 roots and ends pointing in opposite directions. The degree also names the polynomial: linear, quadratic, cubic and so on.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-SSE.A.1.a, CCSS HSA-APR.A.1
Cite this entry
Effortless Math Team. “Degree of a Polynomial.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/degree-of-a-polynomial/
Flashcard
TermDegree of a Polynomialthe highest power
DefinitionThe highest degree of any term (add exponents in a term). 4x3 − 2x5 + 7x has degree 5; 3x2y4 has degree 6.
DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2Leading CoefficientThe leading coefficient of a polynomial is the number multiplying the term with the highest power of the variable. In 3x2 − 5x4 + 2x, the highest power is x4, so the leading coefficient is −5. Write the polynomial in standard form first; its sign tells which way the graph’s ends point.Algebra · Algebra 1, Algebra 2End BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, PrecalculusStandard FormStandard form of a number is the usual way of writing it with digits, such as 4,306, with each digit in its place-value position. It is the short form you get from expanded form (4,000 + 300 + 6) or from word form (four thousand, three hundred six). Zeros hold any empty places.Number System · Grade 2, Grade 3, Grade 4PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2MonomialA monomial is a single term made of a number, a variable, or a product of numbers and variables with whole-number exponents, such as 8, x, or −7x2y. It has no addition or subtraction, and no variable in a denominator. Its degree is the sum of the exponents on its variables.Algebra · Grade 8, Algebra 1BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2TrinomialA trinomial is a polynomial with exactly three terms, such as x2 + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax2 + bx + c, which students learn to factor into two binomial factors.Algebra · Algebra 1, Algebra 2SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6TermA term is a single number, a variable, or numbers and variables multiplied together, such as 7, x or −4xy2. In an expression, terms are the parts separated by plus and minus signs, so 5x2 − 3x + 8 has three terms. The sign in front of a term belongs to it.Algebra · Grade 6, Grade 7, Algebra 1Quadratic EquationA quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2