Calculator inputs
Use expressions of the form a x3n + b x2n + c xn + d. Set a = 0 for a quadratic substitution case.
Example data table
| a | b | c | d | n | Expression | Expected factorization |
|---|---|---|---|---|---|---|
| 1 | -6 | 11 | -6 | 2 | x6 - 6x4 + 11x2 - 6 | (x2 - 1)(x2 - 2)(x2 - 3) |
| 0 | 1 | -5 | 6 | 2 | x4 - 5x2 + 6 | (x2 - 2)(x2 - 3) |
| 0 | 1 | 2 | 1 | 3 | x6 + 2x3 + 1 | (x3 + 1)2 |
Formula used
Start with P(x) = a x3n + b x2n + c xn + d. Let u = xn. Then solve or factor Q(u) = a u3 + b u2 + c u + d. After factoring Q(u), replace u with xn.
For a quadratic substitution case, set a = 0. The tool then applies the quadratic formula to Q(u) = b u2 + c u + d and back-substitutes the roots into x.
How to use this calculator
- Enter coefficients matching a x3n + b x2n + c xn + d.
- Choose n so every repeated exponent is a multiple of n.
- Set graph limits to inspect the original expression visually.
- Press the factor button to view steps, factors, roots, and the curve.
- Use the export buttons to save a compact results report.
Frequently asked questions
1. What does factoring by substitution mean?
It replaces a repeated power pattern with a simpler variable. For example, letting u = x2 changes x4 - 5x2 + 6 into u2 - 5u + 6, which is easier to factor.
2. Which expressions fit this calculator best?
It is designed for expressions that become linear, quadratic, or cubic after setting u = xn. Typical examples include quartics, sextics, and other repeated-exponent polynomials with a common step size.
3. Why do I need to choose n carefully?
n must match the repeated exponent pattern. If the powers are 6, 4, and 2, choose n = 2. If the powers are 9, 6, and 3, choose n = 3. A correct choice creates a simpler polynomial in u.
4. Does the calculator always return exact factors?
No. Exact rational factors appear when the transformed polynomial in u factors neatly. If not, the calculator still shows approximate roots and an approximate factorization so you can continue analysis.
5. What do the u-roots tell me?
The u-roots solve Q(u) = 0 after substitution. Each real u-root may produce one or more real x-roots, depending on n. Even powers need nonnegative u-values for real x-roots.
6. Why can real u-roots give no real x-roots?
When n is even, xn cannot equal a negative real number. So a negative real u-root does not generate a real x-root, even though it still matters algebraically in the factorization.
7. What does the graph represent?
The graph shows the original expression P(x), not the transformed polynomial Q(u). This helps you inspect intercepts, turning behavior, and overall shape across the chosen x-range.
8. When should I use the export buttons?
Use them when you want to keep a quick record of inputs, factorization, roots, and notes. CSV works well for tables, while PDF is useful for sharing a compact printable summary.