Degree and Leading Coefficient Calculator
This degree and leading coefficient calculator reads a polynomial in whatever order it is written and shows you every term with its own degree rather than the bare answer.
Type 5x^7 + 2x^5 - 4x^3 + x^2 + 15 and you get degree 7 with leading coefficient 5.
The Two Definitions, Which Are Short
A polynomial's degree is the highest power of the variable anywhere in the expression. The leading term is the term carrying that power, and the leading coefficient is the number in front of it.
That is genuinely all of it. Which raises a fair question: why does anyone get this wrong?
Because of the Order
Look at 3 + 2x⁵.
Its leading coefficient is 2. Not 3 — even though 3 is written first, and even though it is the number your eye lands on. A university textbook introduces this case with the words "watch out for this trick", which tells you how reliably it catches people.
Polynomials are usually written highest power first, so most of the time the first number you see is the right one. Exam questions know that, and they write them out of order on purpose. This page flags it every time the leading term is not the one you typed first.
| Polynomial | Degree | Leading coefficient |
|---|---|---|
| 2x³ + x + 4 | 3 | 2 |
| 3 + 2x⁵ | 5 | 2 |
| −x⁵ + x⁴ + 2x³ − 1 | 5 | −1 |
| 7 | 0 | 7 |
More Than One Letter Changes the Rule
Here is where a good many students lose a mark they never see coming.
With one variable, the degree of a term is the power of x. With several, it is the sum of the exponents. So −8a²b³c⁴ is not degree 4 — it is degree 9, because 2 + 3 + 4 = 9. And 3xy² is degree 3, since the x carries an invisible exponent of 1.
Type either into the calculator above and it lists each term with its own degree, so the addition is visible rather than assumed.
When There Is No Leading Coefficient at All
Take x² + xy + y². Every term is degree 2. So which one leads?
None of them, without more information. The leading coefficient of a multivariable polynomial depends on an ordering you have to choose — lexicographic, say, or graded — and the question has to give you one. There is no fact of the matter until it does.
This page says so. Several well-known calculators simply pick a term and print its coefficient, which quietly hands you an answer that depends on a decision they made and did not mention.
Is It Even a Polynomial?
3/x + 2 has no degree. Neither does √x + 3, nor x^(1/2) + 1.
A polynomial takes whole-number powers of zero or more, and nothing else. Put a letter underneath a division and you have written x⁻¹; take its square root and you have written x^½. Both are perfectly good algebraic expressions and neither is a polynomial, so neither has a degree to report.
Every calculator I could find will accept these and print a number anyway. That number is a wrong answer wearing a right one's clothes, and it is the kind a student has no way of catching. This page refuses and explains which rule was broken.
The Zero Polynomial
Type x² − x² and everything cancels.
Its degree is undefined — not zero. A constant like 7 has degree 0 because you can write it as 7x⁰. Zero cannot be written that way: 0x⁰ is zero, and so is 0x⁵, and so is 0x¹⁰⁰, so there is no power to single out. Some books assign it −∞ to keep the addition rules working; most simply leave it undefined and move on.
What the Degree Is Good For
Calling it a label undersells it. This is the first thing you find out about a polynomial, and it decides most of what follows.
A degree-n polynomial has at most n roots and at most n − 1 turning points. Both are ceilings rather than counts — a cubic can cross the axis once or three times, but never four. The degree also gives the polynomial its name: constant, linear, quadratic, cubic, quartic, quintic, and then nothing standard, which is why you will see "degree 7" written out in full.
Degree and leading coefficient together also settle what the graph does far out to the left and right, which the end behavior calculator works through properly. If you want the roots themselves, the zeros of polynomial calculator finds them.
Monic Polynomials
A polynomial whose leading coefficient is 1 has a name: monic.
x² − 5x + 6 is monic; 2x² − 10x + 12 is not, even though it has the same roots. That distinction matters more than it looks. A great deal of polynomial theory is stated for monic polynomials only, because dividing through by the leading coefficient is always possible and makes the statements cleaner. This page tells you when yours is one.
How to Use This Degree and Leading Coefficient Calculator
Type the polynomial as written: 5x^7 + 2x^5 - 4x^3 + x^2 + 15. The order does not matter. Powers use ^, coefficients may be fractions or decimals, and an = 0 on the end is simply ignored, since a degree belongs to the expression rather than the equation.
Only x and y sit on the pad; for other letters tap My keyboard. To see the terms rearranged with the highest power first — which is a job of its own — there is a standard form of a polynomial calculator, and for arithmetic on polynomials generally there is the polynomial calculator.
Degree and Leading Coefficient Calculator FAQ
What is the degree of a polynomial?
What is the leading coefficient?
Does the order the polynomial is written in matter?
What is the degree of a term with several letters?
Does the coefficient affect the degree?
What is the degree of a constant like 7?
Is 3/x + 2 a polynomial?
What does monic mean?
Find the largest term degree, take the coefficient of that term with its sign, and check the expression was a polynomial before you start. The order it was written in is the trap; everything else follows from the exponents.
