Calculator Form
Enter coefficients for a quartic, cubic, quadratic, linear, or constant polynomial. Unused leading coefficients can stay as zero.
Formula Used
P(x) = anxn + an-1xn-1 + ... + a1x + a0
G = gcd(an, an-1, ..., a0)
P(x) = G × Q(x)
If r = p / q is a rational root, then p divides the constant term and q divides the leading coefficient.
Once a root is found, divide the polynomial by its matching linear factor and continue factoring the quotient.
D = b2 - 4ac
If D is a positive perfect square, the quadratic splits into rational linear factors. If D is non-square, the quadratic may stay grouped.
How to Use This Calculator
- Enter coefficients from x4 down to the constant term.
- Leave unused leading terms as zero for lower-degree expressions.
- Pick a variable symbol if you want something other than x.
- Set the graph interval and the number of sample points.
- Click Factor Expression to place the result above the form.
- Review the primitive expression, common factor, roots, and notes.
- Use the CSV or PDF buttons to save the current result.
- Check the graph to verify zeros, symmetry, and turning behavior.
Example Data Table
| Expression | Coefficient Set | Factored Result |
|---|---|---|
| x2 - 5x + 6 | [0, 0, 1, -5, 6] | (x - 2)(x - 3) |
| 2x2 + 7x + 3 | [0, 0, 2, 7, 3] | (2x + 1)(x + 3) |
| x3 - 6x2 + 11x - 6 | [0, 1, -6, 11, -6] | (x - 1)(x - 2)(x - 3) |
| x4 - 5x2 + 4 | [1, 0, -5, 0, 4] | (x - 2)(x - 1)(x + 1)(x + 2) |
| 3x3 + 6x2 - 3x - 6 | [0, 3, 6, -3, -6] | 3(x + 2)(x - 1)(x + 1) |
FAQs
1. What kinds of expressions can this calculator factor?
It handles constant, linear, quadratic, cubic, and quartic polynomials entered through coefficients. It is especially useful for expressions that contain common factors, rational roots, repeated roots, grouped terms, or classic polynomial patterns.
2. Does it accept decimal coefficients?
Yes. The calculator scales decimal coefficients into exact integer form before testing rational roots. That keeps the factoring process reliable while still reporting the original expression and its overall extracted factor correctly.
3. Why does one factor sometimes stay grouped?
A grouped remainder means the remaining polynomial does not split into rational linear factors. For example, some quadratics have irrational or complex roots, and some higher-degree remainders do not factor further by rational-root testing.
4. What does the common factor mean?
The common factor is the numerical value removed from every term before deeper factoring starts. Pulling it out first simplifies the polynomial, reduces arithmetic clutter, and makes later pattern checks and rational-root tests easier.
5. Are the displayed roots always all roots?
The calculator lists detected real roots from exact factors and from a remaining quadratic when possible. Complex roots are not shown here, and a grouped cubic or quartic remainder may still hide additional non-rational roots.
6. How should I enter a lower-degree polynomial?
Leave unused leading coefficients as zero. For example, enter 0 for x4 and x3 when working with a quadratic. The calculator automatically removes leading zeros before determining the true degree.
7. What is the graph used for?
The graph helps you visually confirm the expression. X-intercepts often align with real roots, while end behavior and turning points help you judge whether the factored form matches the original polynomial over the chosen interval.
8. Can I save the result for reports or homework notes?
Yes. The CSV export gives a compact text record for spreadsheets or logs, and the PDF export creates a neat summary you can share, print, or attach to coursework, audits, or revision material.