Rational Root Theorem Worksheet + Rational Zeros Cheat Sheet
Practice generating possible rational zeros, testing candidates with synthetic division, factoring confirmed roots, and finding the remaining zeros. Includes 18 problems, a one-page p/q cheat sheet, and a complete answer key.
Browse our collection of free polynomial worksheets and answer keys for operations, factoring, division, graphing, and zeros.
Free for classroom, tutoring, homework, and review use.
Rational Root Theorem Cheat Sheet: p/q → Test → Divide → Finish
This quick reference turns the Rational Root Theorem into a repeatable solving routine. The theorem does not tell you which candidates are roots; it narrows the search to a finite list of rational numbers worth testing.
Download the one-page printable cheat sheet →
Here is an example polynomial with the constant term p and leading coefficient q highlighted so students can see exactly where each part of the theorem comes from.
Try the Rational Root Theorem p/q Generator
Enter a polynomial with integer coefficients. The generator identifies the constant-term factor set p, the leading-coefficient factor set q, and every unique possible rational zero from ±p/q. It does not test which candidates are actual roots.
Examples: 2x^3 - x^2 - 8x + 4 or x⁴ - 5x³ + 5x² + 5x - 6. You may include “= 0”.
Next step: test candidates with substitution or synthetic division. A remainder of 0 confirms a root.
Worked example
f(x) = 2x³ - 5x² - 4x + 3
p factors: ±1, ±3. q factors: ±1, ±2. Possible rational zeros: ±1, ±3, ±1/2, ±3/2.
Testing x = 1/2 gives remainder 0, so 2x - 1 is a factor. The quotient is 2x² - 4x - 6 = 2(x - 3)(x + 1).
All zeros: -1, 1/2, 3.
Three Levels of Rational Root Theorem Practice
Level 1: Generate Candidates
Students identify p and q and list unique possible rational zeros without testing them yet.
Level 2: Test Rational Zeros
Students generate candidates, use synthetic division, and factor polynomials completely over the rationals.
Level 3: Finish All Zeros
Students find rational roots first, divide them out, and finish the remaining quadratic even when its zeros are irrational or complex.
Rational Root Theorem Worksheet Questions
Rational Root Theorem Worksheet Answer Key
The table below exposes the key answers directly on the page. The printable answer-key PDF is easier to use for grading and includes the same progression.
| # | Candidate list or polynomial | Factorization / candidate set | Answer |
|---|---|---|---|
| 1 | x^3 - 6x^2 + 11x - 6 | +/- 1, +/- 2, +/- 3, +/- 6 | Possible rational zeros only |
| 2 | 2x^3 + 5x^2 - x - 6 | +/- 1/2, +/- 1, +/- 3/2, +/- 2, +/- 3, +/- 6 | Possible rational zeros only |
| 3 | 3x^4 - 5x^2 + 8x - 4 | +/- 1/3, +/- 2/3, +/- 1, +/- 4/3, +/- 2, +/- 4 | Possible rational zeros only |
| 4 | 4x^3 - 7x + 10 | +/- 1/4, +/- 1/2, +/- 1, +/- 5/4, +/- 2, +/- 5/2, +/- 5, +/- 10 | Possible rational zeros only |
| 5 | 5x^4 - 2x^3 + 7x - 3 | +/- 1/5, +/- 3/5, +/- 1, +/- 3 | Possible rational zeros only |
| 6 | 6x^5 + x^2 - 12 | +/- 1/6, +/- 1/3, +/- 1/2, +/- 2/3, +/- 1, +/- 4/3, +/- 3/2, +/- 2, +/- 3, +/- 4, +/- 6, +/- 12 | Possible rational zeros only |
| 7 | +/- 1, +/- 2, +/- 3, +/- 6 | (x - 1)(x - 2)(x - 3) | 1, 2, 3 |
| 8 | +/- 1/2, +/- 1, +/- 2, +/- 4 | (x + 2)(2x - 1)(x - 2) | -2, 1/2, 2 |
| 9 | +/- 1/3, +/- 2/3, +/- 1, +/- 4/3, +/- 2, +/- 4 | (x + 2)(3x + 1)(x - 2) | -2, -1/3, 2 |
| 10 | +/- 1/2, +/- 1, +/- 3/2, +/- 3, +/- 9/2, +/- 9 | (x + 3)(2x + 1)(x - 3) | -3, -1/2, 3 |
| 11 | +/- 1/2, +/- 1, +/- 2 | (x + 1)(2x - 1)(x - 2) | -1, 1/2, 2 |
| 12 | +/- 1, +/- 2, +/- 3, +/- 6 | (x + 1)(x - 1)(x - 2)(x - 3) | -1, 1, 2, 3 |
| 13 | +/- 1/2, +/- 1, +/- 3/2, +/- 3 | (x + 1)(2x - 1)(x - 3) | -1, 1/2, 3 |
| 14 | +/- 1/3, +/- 2/3, +/- 1, +/- 2, +/- 3, +/- 6 | (x + 1)(3x - 2)(x - 3) | -1, 2/3, 3 |
| 15 | +/- 1, +/- 2 | (x + 1)(x - 2)(x^2 + 1) | -1, 2, -i, i |
| 16 | +/- 1/2, +/- 1, +/- 3/2, +/- 2, +/- 3, +/- 6 | (x + 3)(2x - 1)(x^2 - 2) | -3, 1/2, -sqrt(2), sqrt(2) |
| 17 | +/- 1, +/- 2, +/- 5, +/- 10 | (x + 2)(x - 1)(x^2 - 5) | -2, 1, -sqrt(5), sqrt(5) |
| 18 | +/- 1/2, +/- 1, +/- 2, +/- 4 | (x + 1)(2x - 1)(x^2 + 4) | -1, 1/2, -2i, 2i |
Common Rational Root Theorem Mistakes
Reversing p and q
p comes from the constant term; q comes from the leading coefficient.
Forgetting ±
Both positive and negative ratios must be considered unless another theorem has already ruled them out.
Treating candidates as answers
A p/q value is only a possible zero until substitution or synthetic division gives remainder 0.
Keeping duplicate fractions
Reduce ratios such as 2/2 to 1 and list each unique candidate only once.
Stopping after one root
Divide out the confirmed factor and keep solving the lower-degree quotient.
Forgetting non-rational zeros
The theorem only generates rational candidates. A final quadratic can still have irrational or complex zeros.
Related Polynomial Worksheets and References
Additional free education references
Created and reviewed by Berke Sahbazoglu
Owner and lead math tutor, Burke Tutoring in Fremont. 10+ years tutoring Algebra I & II. These problems were adapted from Rational Root Theorem practice I use with Algebra II students and independently checked before publication. This resource was reviewed for candidate generation, synthetic-division remainders, factorization, and final zero sets.