Factoring Higher-Degree Generator
The Factoring Higher-Degree Polynomials lesson covers sum and difference of cubes, grouping, and quadratic-in-form substitution in full; this generator drills the result of each, asking for one specific constant from the factored form.
How the difficulty levels work
| Difficulty | What it drills | Example |
|---|---|---|
| Easy | Sum or difference of cubes, find \(b\) | \(x^{3}-125=(x-5)(x^{2}+5x+25)\), \(b=5\) |
| Medium | The same, with a GCF pulled out first | Find the GCF or \(b\) |
| Hard | A quadratic-in-form quartic, or a bigger cubes problem | Four linear roots: \(\pm p\), \(\pm q\) |
Easy factors \(x^{3}\pm b^{3}\) directly, asking for \(b\), the constant in the binomial factor.
Medium adds a GCF in front of the cube, alternating between asking for that GCF and asking for \(b\) — two separate, equally necessary numbers.
Hard alternates between a quadratic-in-form quartic that factors into four linear roots (asking for the larger one) and a bigger sum-or-difference-of-cubes problem.
Using the generator
Pick a difficulty and a question count, and a fresh set appears instantly. Every answer here is a whole number — enter it exactly as computed.
- Check Answers scores the set and flags exactly which problems need another look
- Show Answers reveals every solution, useful for reviewing a paper attempt
- Print Worksheet outputs a clean page for offline practice or classroom handouts
The rules this generator drills
| Rule | Statement |
|---|---|
| Difference of cubes | \(a^{3}-b^{3} = (a-b)(a^{2}+ab+b^{2})\) |
| Sum of cubes | \(a^{3}+b^{3} = (a+b)(a^{2}-ab+b^{2})\) |
| Quadratic in form | Substitute \(u=x^{2}\), factor as a quadratic, then substitute back |
See the Factoring Higher-Degree Polynomials lesson for the full technique and worked examples combining several steps.
Common mistakes to watch for
Mixing up the signs in the cubes patterns. The trinomial factor’s middle sign is always the opposite of the binomial’s, and its last term is always positive.
Forgetting to substitute back after quadratic-in-form factoring. The substitution variable is only a bookkeeping tool — the final answer is written in terms of the original variable.
Skipping the GCF check between steps. A GCF can appear inside a factor that wasn’t obvious until after an earlier factoring step already ran.
Stopping after the first factoring step. A quartic factored into two quadratics may still have more factoring left — check every resulting factor again.
Where to go next
For the trinomial techniques this lesson builds on, see the Factoring Quadratics Generator and the Factoring by Grouping Generator. For the full technique behind every problem here, see the Factoring Higher-Degree Polynomials lesson.