Calculator
Enter polynomial coefficients from highest degree to constant. Example: 3,-2,5,-7 means 3x³ - 2x² + 5x - 7.
Export Results
Download the current calculation as CSV or PDF.
| Metric | Value |
|---|---|
| Operation | Multiply polynomial by monomial |
| Polynomial | 3x^3 - 2x^2 + 5x - 7 |
| Monomial | 2x^3 |
| Result Expression | 6x^6 - 4x^5 + 10x^4 - 14x^3 |
| Result Degree | 6 |
| Result at x=2 | 272 |
| Derivative of Polynomial | 9x^2 - 4x + 5 |
| Note | Submit the form to replace this example with live output. |
Plotly Graph
The chart compares the original polynomial with the transformed result whenever the output is a polynomial expression.
Example Data Table
| Example | Polynomial Coefficients | Monomial | Operation | Sample Result |
|---|---|---|---|---|
| 1 | 3,-2,5,-7 | 2x^3 | Multiply | 6x^6 - 4x^5 + 10x^4 - 14x^3 |
| 2 | 1,0,-4,9 | 5x^2 | Add | x^3 + 5x^2 - 4x + 9 |
| 3 | 8,4,0,-12 | 2x | Divide | 4x^2 + 2x - 6/x |
| 4 | 6,-3,2 | 4x | Subtract | 6x^2 - 7x + 2 |
Formula Used
Polynomial Form
P(x) = anxn + an-1xn-1 + ... + a1x + a0
Monomial Form
M(x) = c xm
Multiplication Rule
P(x) × M(x) multiplies every coefficient by c and increases each exponent by m.
Addition and Subtraction Rule
The monomial combines only with the polynomial term having the same exponent. All other terms remain unchanged.
Division Rule
Each term is divided by c xm. Exponents decrease by m. Negative exponents are shown in term-wise form.
How to Use This Calculator
- Enter the polynomial coefficients in descending degree order.
- Choose a variable symbol such as x or y.
- Select whether you want to multiply, divide, add, or subtract.
- Enter the monomial coefficient and exponent.
- Add an evaluation value to compare numeric outputs.
- Set the graph range for the Plotly chart.
- Click Calculate Now to show the result above the form.
- Use the CSV and PDF buttons to export your output.
Frequently Asked Questions
1. What is a monomial?
A monomial is a single algebraic term, such as 4x² or -7y. It contains a coefficient and a nonnegative integer exponent.
2. What is a polynomial?
A polynomial is a sum of monomials. Examples include 3x³ - 2x + 5 and x² + 4x + 9.
3. How are coefficients entered here?
Enter coefficients from the highest exponent to the constant term. For 2x² - 3x + 7, type 2,-3,7.
4. Can this calculator divide every polynomial perfectly?
Not always. If dividing by a monomial creates negative exponents, the calculator shows a term-wise expression instead of a standard polynomial quotient.
5. Why is the graph missing the result curve sometimes?
The result curve appears when the output remains a polynomial. Non-polynomial term-wise division results are not plotted as a standard polynomial line.
6. What does the derivative section show?
It shows the derivative of the original polynomial. This helps you inspect slope behavior and adds extra algebra insight.
7. Can I use decimals and negative numbers?
Yes. The calculator accepts decimal coefficients, negative values, and negative evaluation points for flexible algebra practice.
8. What do the export buttons save?
The CSV button saves the visible result table. The PDF button creates a simple report with the same calculation summary.