Calculator inputs
This tool extracts common factors, tests rational roots, applies synthetic division, and graphs the polynomial on your selected interval.
Example data table
| Polynomial | Factored form | Key note |
|---|---|---|
| x^2 - 5x + 6 | (x - 2)(x - 3) | Two distinct rational roots. |
| 2x^2 + 7x + 3 | (2x + 1)(x + 3) | Non-monic quadratic with rational roots. |
| x^3 - 6x^2 + 11x - 6 | (x - 1)(x - 2)(x - 3) | Three simple real roots. |
| x^4 + 5x^2 + 4 | (x^2 + 1)(x^2 + 4) | Even quartic treated as a quadratic in x^2. |
| 3x^4 - 12x^3 + 12x^2 | 3(x)^2(x - 2)^2 | GCF extraction and repeated roots. |
Formula used
1. Greatest common factor: pull out g = gcd(a_n, a_{n-1}, ..., a_0) before testing special structures.
2. Rational Root Theorem: if p/q is a rational root in lowest terms, then p divides the constant term and q divides the leading coefficient.
3. Synthetic division: once a root is found, divide the polynomial by (x - r) to reduce its degree and continue factoring.
4. Even quartic pattern: if only even powers appear, rewrite ax^4 + bx^2 + c as ay^2 + by + c with y = x^2.
5. Numerical roots: approximate roots are computed iteratively so the graph and root list still help when exact rational factoring stops.
How to use this calculator
- Select the degree of your polynomial.
- Enter each coefficient from the highest power down to the constant term.
- Set the graph interval and the number of plot points.
- Press Factor Polynomial to generate the exact factorization summary above the form.
- Review the steps, approximate roots, and graph to validate the result.
- Use the CSV or PDF buttons to save the result for later reference.
FAQs
1. What polynomials can this tool factor exactly?
It handles integer-coefficient polynomials up to degree six here, extracts common factors, finds rational roots, reduces degree with synthetic division, and keeps any rational-irreducible remainder visible.
2. Why is a polynomial sometimes only partially factored?
Some expressions contain irrational, complex, or higher-order irreducible components. In those cases, the tool shows the exact rational part it can prove and leaves the remaining factor intact.
3. Does the tool identify repeated roots?
Yes. Repeated linear factors appear more than once during synthetic division, and the rational-root summary reports multiplicity whenever the same exact root is detected multiple times.
4. Why should I enter integer coefficients?
Exact rational-root testing relies on divisors of the constant and leading coefficients. Integer input keeps those divisors meaningful and prevents rounding noise from corrupting exact factor detection.
5. What does the graph add to the factoring result?
The graph helps you confirm x-intercepts, turning behavior, repeated-root flattening, and sign changes. It is especially useful when a polynomial is only partially factored exactly.
6. What do the CSV and PDF buttons export?
They export the current result summary. The CSV stores structured values, while the PDF saves a readable text version of the same output for quick sharing or documentation.
7. Is synthetic division the same as long division?
They produce the same quotient and remainder when dividing by a linear factor. Synthetic division is simply a faster tabular shortcut for that special situation.
8. Does a graph alone prove the factorization?
No. The graph is visual evidence, not a proof. Exact factors come from algebraic tests such as the GCF step, Rational Root Theorem, and synthetic division.