Degree of Polynomial Calculator & How It Works
The calculator for this page is switched off while it is checked. The formula is below, and the worked example in the article shows how to use it by hand.
This calculator finds the degree of a polynomial you type in, returning the highest power of the variable after combining like terms. You enter a polynomial such as 3x^4 - 2x^2 + x - 7, and it outputs the degree (4 in this case) along with the leading coefficient and the number of terms. The degree is the single most important number for understanding a polynomial’s end behavior, number of roots, and graphing shape.
Formula
For a polynomial in one variable written in standard form
anxn + an-1xn-1 + … + a1x + a0
the degree is n, the largest exponent with a nonzero coefficient. If the polynomial has more than one variable, the degree is the largest sum of exponents in any single term. For example, 5x2y3 has degree 2+3 = 5. The leading coefficient is an, the coefficient of the term with the highest degree. A constant like 7 has degree 0, and the zero polynomial (0) is usually said to have no degree or degree undefined.
Worked Example
Using the default polynomial 3x^4 - 2x^2 + x - 7:
- Write each term with its exponent:
3x4(exponent 4),-2x2(exponent 2),x(exponent 1, since x = x1),-7(exponent 0, since -7 = -7x0). - Find the largest exponent: 4.
- The degree is 4.
- The leading coefficient is the coefficient of that term: 3.
- The number of terms is 4.
So the calculator shows degree = 4, leading coefficient = 3, and number of terms = 4.
Why the Degree Matters
The degree tells you the maximum number of roots (solutions) the polynomial can have. A degree 4 polynomial can have up to 4 real roots, but it might have fewer. It also determines the end behavior: for even degrees, both ends of the graph go the same direction (up if the leading coefficient is positive, down if negative); for odd degrees, the ends go in opposite directions. The degree also tells you the maximum number of turning points (local maxima and minima) is degree minus 1. For example, a cubic (degree 3) has at most 2 turning points.
How to Enter a Polynomial
Use ^ for exponents, * for multiplication (optional), and parentheses for grouping. The calculator accepts standard notation:
2x^3 - 5x + 1– validx^2 + 2x + 1– valid3x2– invalid (needs ^)4x^2y^3– valid, but degree is 5 (2+3)
If you omit the caret, the calculator may misinterpret x2 as a variable name. Always use ^ for clarity.
Common Mistakes and Misconceptions
A frequent error is forgetting to combine like terms first. For example, x^2 + 3x^2 simplifies to 4x^2, so the degree is 2, not 4. Another mistake: treating a coefficient of zero as a term. 0x^5 + x^2 has degree 2, not 5, because the x^5 term vanishes. Also, remember that a constant like 5 has degree 0, not 1. Some people think x has degree 0, but it is degree 1. Finally, for multivariable polynomials, you must add the exponents in each term, not just look at the first variable.
Limits of the Calculator
This calculator handles polynomials with integer, fractional, or decimal coefficients. It does not handle trigonometric functions (sin x), logarithms, or exponential terms – those are not polynomials. It also does not simplify the polynomial for you beyond combining like terms; if you enter x^2 + 2x + 1, it will give degree 2, but it won’t factor it. For very large exponents (above 1000), the calculator may round or fail due to numerical limits, but such polynomials are rare in practice.
Frequently Asked Questions
What is the degree of a polynomial with no variable?
A constant like 7 or -3 has degree 0, because it can be written as 7x0. The zero polynomial (0) is special – it has no degree, or sometimes is said to have degree -∞, but most textbooks simply say it is undefined.
How do I find the degree of a polynomial with multiple variables?
For each term, add the exponents of all variables in that term. The degree of the polynomial is the largest such sum. Example: 3x^2y^3 + xy – the first term has degree 2+3=5, the second has degree 1+1=2, so the polynomial has degree 5.
What is the degree of a linear polynomial?
A linear polynomial has degree 1. It is of the form ax + b, where a ≠ 0. Examples: 2x+3, -x+5, 0.5x-1.
Can a polynomial have a negative degree?
No. By definition, a polynomial has nonnegative integer exponents. Expressions with negative exponents, like x-2, are not polynomials, so they don’t have a polynomial degree.
Does the degree change if I multiply two polynomials?
Yes. The degree of the product is the sum of the degrees of the two polynomials, provided the leading coefficients are nonzero. For example, (x^2 + 1)(x^3 + x) has degree 2+3=5.
What is a monomial, binomial, trinomial?
These names refer to the number of terms: monomial (1 term), binomial (2 terms), trinomial (3 terms). They are not directly related to degree, but a monomial like 5x^3 has degree 3, while a binomial like x^2 + 1 has degree 2.