Calculator input
Use expressions such as 2x^4 - 3x^3 + x - 7 and x^2 - 2.
Formula used
For nonzero divisor D(x), polynomial division writes the dividend as:
P(x) = D(x)Q(x) + R(x)
Here, Q(x) is the quotient. R(x) is the remainder. The degree of R(x) must be smaller than the degree of D(x).
At each step, the calculator divides the leading term of the current remainder by the leading term of the divisor. It multiplies the full divisor by that term. Then it subtracts the product and repeats.
How to use this calculator
- Enter the dividend polynomial in the first box.
- Enter the nonzero divisor polynomial in the second box.
- Set decimal precision, graph range, plot points, and tolerance.
- Press the calculate button to view quotient and remainder.
- Review the step table and identity check.
- Use CSV or PDF buttons to save the report.
Example data table
| Dividend | Divisor | Quotient | Remainder |
|---|---|---|---|
| 2x^4 - 3x^3 + x - 7 | x^2 - 2 | 2x^2 - 3x + 4 | -5x + 1 |
| x^3 - 6x^2 + 11x - 6 | x - 1 | x^2 - 5x + 6 | 0 |
| 4x^5 + 2x^3 - x + 8 | 2x^2 + 1 | 2x^3 | -x + 8 |
Polynomial Long Division Basics
Polynomial long division separates one polynomial into a quotient and a remainder. It works like number division, but each step uses the leading term. The method is useful when factoring, simplifying rational expressions, checking roots, and preparing graphs. It also helps students see how powers decrease during division.
Why This Calculator Helps
Manual division can be slow when degrees are high or coefficients are decimal values. This calculator accepts a dividend and divisor, then finds each quotient term. It subtracts the shifted divisor at every stage. The result shows the quotient, remainder, identity check, and value check at a selected x value. These details make the answer easier to audit.
Using Results Correctly
The main identity is dividend equals divisor times quotient plus remainder. A small remainder may appear when decimal coefficients create rounding noise. Use the precision setting to control displayed values. The graph is not a proof, but it is useful for comparison. When the dividend curve matches the rebuilt curve, the division is likely consistent.
Learning and Practice
For practice, start with a divisor such as x - 2. Then try a quadratic divisor. Watch the step table. Each row shows which leading terms were matched. This builds a clear habit. First divide leading terms. Next multiply the full divisor. Then subtract. Finally repeat until the remainder degree is lower than the divisor degree.
Advanced Checks
A strong answer should pass several checks. The quotient degree should equal the dividend degree minus the divisor degree when division is possible. The remainder degree should be smaller than the divisor degree. The evaluation check should show almost zero difference after rebuilding the expression. If not, review the input format and signs.
Common Input Tips
Write powers with the caret symbol. Use x^3, not x3. Include missing terms only when helpful, such as 0x^2. Spaces are allowed. Multiplication marks are optional. For example, 3*x^2 and 3x^2 both work. Avoid negative exponents, because they are not polynomial terms.
Export and Review
Download the CSV when you need spreadsheet notes. Download the PDF when you need a printable record. Save both after changing values, precision, or graph range later.
FAQs
1. What is polynomial long division?
Polynomial long division is a method for dividing one polynomial by another. It returns a quotient and a remainder. It works by matching leading terms, multiplying the divisor, subtracting, and repeating.
2. What input format should I use?
Use x as the variable. Write powers with a caret, such as x^4. You may use spaces and multiplication marks. Decimals and negative coefficients are supported.
3. When is the remainder zero?
The remainder is zero when the divisor divides the dividend exactly. In that case, the quotient multiplied by the divisor rebuilds the dividend without any leftover polynomial.
4. Can I divide by a constant?
Yes. A nonzero constant divisor is allowed. The calculator divides every coefficient in the dividend by that constant and gives a zero remainder.
5. Why do I see tiny decimal errors?
Small decimal differences can appear because computers store many decimals approximately. Increase precision or adjust tolerance when values are very close to zero.
6. What does the graph show?
The graph plots the dividend, divisor, quotient, and remainder over your selected x range. It helps compare shapes and inspect possible behavior visually.
7. Is the identity check important?
Yes. The identity confirms that Dividend equals Divisor times Quotient plus Remainder. It is the strongest quick check for the division result.
8. Can I export my work?
Yes. Use the CSV button for spreadsheet use. Use the PDF button for a clean printable report with summary and step details.