Calculator Form
Example Data Table
| Expression | Leading term | Overall degree | Type |
|---|---|---|---|
| 7x^5 - 3x^2 + 8 | 7x^5 | 5 | Single-variable polynomial |
| 4x^2y^3 - 6xy + 1 | 4x^2y^3 | 5 | Multivariable polynomial |
| 9 | 9 | 0 | Constant polynomial |
| x^4 + x^3 + x^2 + x + 1 | x^4 | 4 | Complete polynomial |
| 3a^2b + 8ab^2 - 5 | 3a^2b | 3 | Multivariable polynomial |
Formula Used
Term degree rule
For a term like c·xaybzc, the term degree is the sum of all exponents:
deg(term) = a + b + c
Polynomial degree rule
The degree of the whole polynomial is the largest term degree among all nonzero terms:
deg(P) = max(deg(term))
Leading term rule
After simplifying and sorting terms by descending total degree, the first nonzero term is reported as the leading term.
Special cases
A nonzero constant has degree 0.
The zero polynomial has no defined degree, so this tool labels it as undefined.
How to Use This Calculator
- Type your polynomial in expanded form, such as 5x^4 - 2x^2 + 3 or 4x^2y^3 - 7xy + 1.
- Optionally enter evaluation values like x=2, y=1 if you also want the polynomial’s numeric result.
- Set the graph minimum, maximum, and sample count for the Plotly output.
- Press the calculate button to see the degree, leading term, simplified expression, analysis table, and graph above the form.
- Use the CSV and PDF buttons to export your result summary and term details.
FAQs
1) What is a polynomial degree?
It is the highest exponent total among all nonzero terms in a polynomial. It helps describe growth, shape, and dominant behavior.
2) How is degree found in multivariable terms?
Add the exponents inside one term. For example, x^2y^3 has total degree 5 because 2 + 3 = 5.
3) What if several terms share the highest degree?
The polynomial degree stays the same. This tool then reports the first simplified highest-degree term after sorting by total degree and signature.
4) Do constants have a degree?
Yes. Any nonzero constant polynomial has degree 0 because it contains no variable exponents to raise the total degree.
5) What about the zero polynomial?
Its degree is undefined. Since every coefficient is zero, there is no highest nonzero term to measure.
6) Does this support negative or fractional exponents?
No. Those forms are not polynomial terms. This calculator accepts only whole, non-negative integer exponents.
7) Why is the graph sometimes a bar chart?
Multivariable polynomials cannot be shown fully on one simple line. The bar chart instead compares degrees across simplified terms.
8) Can repeated like terms change the result?
Yes. Like terms are combined before the final degree is reported, so cancellations and simplifications can affect the leading term.