Calculator
Enter a polynomial using integer coefficients. Example: 12x^3y + 18x^2y^2 - 6xy
Example Data Table
| Polynomial | GCF | Quotient Polynomial | Factored Form |
|---|---|---|---|
| 12x^3y + 18x^2y^2 - 6xy | 6xy | 2x^2 + 3xy - 1 | 6xy(2x^2 + 3xy - 1) |
| 8a^2b + 12ab^2 | 4ab | 2a + 3b | 4ab(2a + 3b) |
| 15m^3n - 20m^2n^2 + 5mn | 5mn | 3m^2 - 4mn + 1 | 5mn(3m^2 - 4mn + 1) |
| 9x^2 + 27x | 9x | x + 3 | 9x(x + 3) |
Formula Used
General idea: Find the greatest common factor across all coefficients and each shared variable.
Coefficient GCF: GCF(a, b, c, ...)
Variable GCF: Use the smallest exponent for each variable present in every term.
Factoring rule: Polynomial = GCF × Quotient Polynomial
Example: 12x^3y + 18x^2y^2 - 6xy
Coefficient GCF is 6.
Common variable factor is xy.
So the factored form becomes 6xy(2x^2 + 3xy - 1).
How to Use This Calculator
- Enter your polynomial in one input box.
- Use integer coefficients and standard variable notation.
- Write exponents with the caret symbol.
- Click Factor GCF to process the expression.
- Review the GCF, quotient, and final factored form.
- Check the breakdown table for every term.
- Use CSV or PDF export for saving results.
- Study the graph to compare coefficient changes.
FAQs
1. What does GCF mean in polynomial factoring?
GCF means greatest common factor. It is the largest monomial that divides every term in the polynomial without leaving fractions or remainders.
2. Does the calculator handle variables with exponents?
Yes. It checks each variable across all terms and keeps the smallest shared exponent. That shared variable factor becomes part of the final GCF.
3. Should I include spaces in the expression?
Spaces are allowed. The calculator removes extra spacing automatically before parsing the polynomial and factoring the common monomial factor.
4. Can this tool factor the entire polynomial completely?
This version focuses on factoring out the greatest common factor first. Some quotient polynomials may still have other factorization opportunities afterward.
5. Why is the GCF sometimes only a number?
That happens when coefficients share a divisor, but variables do not appear in every term. Then the common factor comes only from the numbers.
6. What input style works best here?
Use terms like 12x^2, -18xy, or 6y^3. Keep coefficients as integers for the clearest and most reliable factoring result.
7. What does the graph show?
The graph compares original absolute coefficients with quotient absolute coefficients. It helps you see how each term changes after removing the GCF.
8. Can I save my factoring steps?
Yes. Use the CSV button for spreadsheet-friendly output or the PDF button for a quick printable report of the current factoring result.