Result area
Submit the form to display the factorization above the calculator and below the header.
Calculator Inputs
Plotly Graph
The graph compares the original expression with its factorized form. For two-variable identities, y stays fixed at 1 while x changes across the selected range.
Example Data Table
| Pattern | Expanded expression | Factored form | Note |
|---|---|---|---|
| Difference of squares | 9x2 - 16y2 | (3x - 4y)(3x + 4y) | Use conjugates. |
| Perfect square plus | 9x2 + 24xy + 16y2 | (3x + 4y)2 | Middle term equals 2ab. |
| Perfect square minus | 25m2 - 30mn + 9n2 | (5m - 3n)2 | Check the negative middle sign. |
| Sum of cubes | 8x3 + 27y3 | (2x + 3y)(4x2 - 6xy + 9y2) | Use plus, minus, plus. |
| Difference of cubes | 64a3 - b3 | (4a - b)(16a2 + 4ab + b2) | Use plus signs in the quadratic factor. |
Formula Used
- Difference of squares: a2 - b2 = (a - b)(a + b)
- Perfect square sum: a2 + 2ab + b2 = (a + b)2
- Perfect square difference: a2 - 2ab + b2 = (a - b)2
- Sum of cubes: a3 + b3 = (a + b)(a2 - ab + b2)
- Difference of cubes: a3 - b3 = (a - b)(a2 + ab + b2)
The calculator maps your coefficients to these identities, checks whether the required pattern exists, and then verifies the result numerically with the sample values you provide.
How to Use This Calculator
- Select the special-product pattern you want to test.
- Enter variable names for the two symbolic terms.
- Fill in A and B. Enter M only for perfect square trinomials.
- Provide sample values for numeric verification.
- Choose the graph range for the first variable.
- Press Factor Special Product to view the result above the form.
- Review the identity, factorization, worked steps, and graph.
- Use the CSV or PDF buttons to export the calculation summary.
FAQs
1. What are special products in algebra?
Special products are expressions that follow standard identities, such as difference of squares, perfect square trinomials, and sums or differences of cubes. Recognizing them speeds up factoring and checking work.
2. Why does the calculator ask for a pattern first?
Each identity has a different structure. Choosing the pattern tells the calculator which formula to test, which signs matter, and how to build the matching factorization steps.
3. What does the middle coefficient mean?
For perfect square trinomials, the middle coefficient must equal 2ab after matching the outer square terms. If it does not, the expression is not a perfect square identity.
4. Can this tool handle irrational square or cube roots?
Yes. When the coefficient is not a perfect square or perfect cube, the calculator displays radicals or cube roots in the factorization so the identity still works over the real numbers.
5. Why are sample values included?
Sample values verify that the original expression and the factorized expression produce the same number. This helps confirm the factorization and supports the graph display.
6. What does the graph represent?
The graph plots the original expression and, when available, its factorized equivalent. The second variable stays fixed, so you can compare both forms as the first variable changes.
7. When would the calculator show a warning instead of a factorization?
A warning appears when your coefficients do not fit the chosen special-product identity, such as an incorrect middle term for a perfect square trinomial.
8. Can I export the result for study notes?
Yes. The CSV export is useful for tabular records, and the PDF export creates a clean summary you can save, print, or attach to lesson materials.