Calculator Inputs
Enter coefficients from highest degree to constant term. Example: 3, -2, 5 means 3x² − 2x + 5.
Example Data Table
| Example | Polynomial A coefficients | Polynomial B coefficients | Expanded product |
|---|---|---|---|
| Quadratic × quadratic | 3, -2, 5 | 1, 4, -1 | 3x^4 + 10x^3 - 6x^2 + 22x - 5 |
| Cubic × linear | 2, 0, -3, 1 | 1, -2 | 2x^4 - 4x^3 - 3x^2 + 7x - 2 |
| Binomial × binomial | 1, -7 | 1, 7 | x^2 - 49 |
| Zero middle terms | 4, 0, 0, -9 | 1, 0, 3 | 4x^5 + 12x^3 - 9x^2 - 27 |
Formula Used
Polynomial multiplication uses coefficient convolution. Each coefficient in the product comes from adding every valid pairwise term product that lands on the same degree.
Degree rule: If neither polynomial is zero, then
Interpretation: The calculator multiplies every term in the first polynomial by every term in the second, groups like powers, and returns the fully expanded result.
How to Use This Calculator
- Enter the coefficients for Polynomial A from highest power to constant.
- Enter the coefficients for Polynomial B in the same order.
- Choose the variable symbol you want displayed in the algebraic output.
- Set the graph range and point count for plotting both inputs and the product.
- Press Multiply Polynomials to show the expanded product above the form.
- Review the summary cards, coefficient table, degree summary, and step-by-step term products.
- Use the CSV button for spreadsheet export or the PDF button for a printable report.
- Check the Plotly graph to compare how the two inputs and their product behave across the selected x-range.
FAQs
1. In what order should I enter coefficients?
Enter them from highest degree to constant term. For example, 3x² − 2x + 5 becomes 3, -2, 5. If a power is missing, include zero in that position.
2. Can I use decimals and negative numbers?
Yes. The calculator accepts integers, decimals, and negative values. That makes it useful for classroom work, numeric modeling, engineering approximations, and any case where polynomial coefficients are not whole numbers.
3. What happens if one polynomial is zero?
If every coefficient in one polynomial is zero, the product is the zero polynomial. The tool normalizes the output and displays a clean result instead of leaving unnecessary leading zero terms.
4. Why does the product degree usually equal the sum of degrees?
The highest-degree term in the product comes from multiplying the two leading terms. Unless one leading coefficient is zero after normalization, the degrees add directly and determine the product degree.
5. Why is a coefficient sometimes missing in the final expression?
That usually means the coefficient became zero after combining like terms. For example, positive and negative contributions can cancel, leaving a missing middle term in the simplified product.
6. What does the graph show?
The graph plots Polynomial A, Polynomial B, and their product across your chosen x-range. It helps you compare roots, turning behavior, magnitude growth, and overall shape after multiplication.
7. What is included in the CSV and PDF downloads?
Both exports include the input expressions, product expression, coefficient lists, degree summary, and detailed multiplication steps. They are useful for homework records, reports, or checking manual calculations later.
8. Can this calculator help verify my hand-worked expansion?
Yes. The step table shows every pairwise term multiplication, and the degree summary shows how terms combine. That makes it easier to spot sign mistakes, missing terms, or degree alignment errors.