Calculator inputs
Use up to six terms. Each enabled card supports one coefficient and three exponents.
Example data table
| Polynomial sample | Order | Priority | Leading term | Reason |
|---|---|---|---|---|
| 5x4y - 3x2y5 + 7xz6 + 2x3y2z | Graded lexicographic | x > y > z | 5x4y | Total degree ties at five, then lex order selects x4y. |
| 4a2b3 + 9ab4 - 2b5 | Lexicographic | a > b > c | 4a2b3 | Lexicographic ranking favors the highest exponent on a first. |
| 6m2n2p + 8mn3p - 5m2p2 | Graded reverse lexicographic | m > n > p | 8mn3p | All degree-four terms tie, so reverse lex breaks the tie. |
Formula used
Leading term is the nonzero term that ranks highest under the selected monomial order.
Total degree: degree = e1 + e2 + e3, where each e is a variable exponent.
General term: T = c × v1e1 × v2e2 × v3e3.
Lexicographic order: compare exponents using the chosen priority, starting from the highest-priority variable.
Graded lexicographic order: compare total degrees first, then use lexicographic comparison to break ties.
Graded reverse lexicographic order: compare total degrees first, then inspect exponents from the lowest-priority variable backward.
How to use this calculator
- Choose three variable symbols, even if some exponents stay zero.
- Select the monomial order that matches your textbook or assignment.
- Set variable priority to control tie-breaking between similar terms.
- Enable each term you want included in the polynomial comparison.
- Enter the coefficient and exponents for every active term card.
- Submit the form to display the leading term above the inputs.
- Use the CSV or PDF buttons to save the ranked results.
FAQs
1. What is a leading term?
A leading term is the highest-ranked nonzero term in a polynomial after applying a chosen monomial order. Different orders can change which term appears first.
2. Why can the leading term change?
It changes because lexicographic, graded lexicographic, and graded reverse lexicographic orders compare exponents differently. The same polynomial can therefore produce different leading terms.
3. Does coefficient size decide the leading term?
No. The ordering mainly depends on exponent patterns and degree. The coefficient stays attached to the selected term, but magnitude alone does not rank terms higher.
4. Can I use one-variable polynomials?
Yes. Enter your main variable, then keep the other exponents at zero. The calculator still evaluates the polynomial correctly using the chosen order.
5. Are negative coefficients allowed?
Yes. Negative coefficients are accepted and preserved in the displayed result. Only the exponent pattern controls ranking, provided the coefficient is not zero.
6. What happens with zero coefficients?
Zero-coefficient terms are ignored because they do not contribute to the polynomial. At least one enabled term must have a nonzero coefficient for a valid result.
7. What is variable priority?
Variable priority defines which exponent gets checked first when a monomial order needs tie-breaking. For example, x > y > z means x is compared before y and z.
8. When is this useful?
It helps with algebra classes, symbolic computation, Gröbner basis preparation, polynomial ordering practice, and quick verification of textbook examples or homework steps.