Descartes' Rule of Signs Worksheet + Root Profile Generator
Practice using Descartes' Rule of Signs to determine possible positive real zeros, possible negative real zeros, and the number of nonreal zeros that may remain. Includes 18 problems, a one-page root-profile cheat sheet, an interactive analyzer, 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 Precalculus review.
Descartes' Rule of Signs Cheat Sheet: Positive → Negative → Nonreal
Descartes' Rule does not find the zeros themselves. It limits how many positive and negative real zeros are possible. Once those possibilities are combined, the degree tells you how many nonreal zeros may remain.
Download the one-page printable cheat sheet →
Worked example: build the whole root profile
f(x) = x⁵ - 3x⁴ - 2x³ + 6x² + 5x - 7
+ - - + + - has 3 sign changes → positive roots: 3 or 1.-x⁵ - 3x⁴ + 2x³ + 6x² - 5x - 7- - + + - - has 2 changes → negative roots: 2 or 0.(3,2), (3,0), (1,2), (1,0).
(3 positive, 2 negative, 0 nonreal); (3 positive, 0 negative, 2 nonreal); (1 positive, 2 negative, 2 nonreal); (1 positive, 0 negative, 4 nonreal).
Descartes' Rule Root Profile Generator
Enter a polynomial with integer coefficients. The analyzer counts sign changes in f(x) and f(-x), lists the possible positive and negative real-zero counts, and builds every valid nonreal-root profile.
Use integer coefficients, degree 1-10, and coefficient magnitudes up to 100. Missing powers are allowed. If the constant term is 0, the generator factors out the zero roots first.
Three Levels of Descartes' Rule Practice
Level 1: Sign Changes & Real Zeros
Count V+ and V- and state the possible positive and negative real-zero counts.
Level 2: Add Nonreal Zeros
Use the degree to determine every possible number of nonreal zeros after the real-root possibilities are known.
Level 3: Full Root Profiles
List every valid positive / negative / nonreal combination, including a challenge with zero roots.
Descartes' Rule of Signs Worksheet Questions
Descartes' Rule of Signs Worksheet Answer Key
The table shows the number of sign changes first, followed by the allowed counts. Descartes' Rule counts multiplicity and gives possibilities rather than exact root values.
| # | V+ → possible positive | V- → possible negative | Nonreal / full profile |
|---|---|---|---|
| 1 | 4 → 4, 2, 0 | 0 → 0 | - |
| 2 | 2 → 2, 0 | 3 → 3, 1 | - |
| 3 | 3 → 3, 1 | 2 → 2, 0 | - |
| 4 | 2 → 2, 0 | 2 → 2, 0 | - |
| 5 | 2 → 2, 0 | 2 → 2, 0 | - |
| 6 | 0 → 0 | 4 → 4, 2, 0 | - |
| 7 | 3 → 3, 1 | 2 → 2, 0 | Possible nonreal: 0, 2, 4 |
| 8 | 4 → 4, 2, 0 | 1 → 1 | Possible nonreal: 0, 2, 4 |
| 9 | 2 → 2, 0 | 2 → 2, 0 | Possible nonreal: 0, 2, 4 |
| 10 | 3 → 3, 1 | 1 → 1 | Possible nonreal: 0, 2 |
| 11 | 6 → 6, 4, 2, 0 | 0 → 0 | Possible nonreal: 0, 2, 4, 6 |
| 12 | 2 → 2, 0 | 3 → 3, 1 | Possible nonreal: 0, 2, 4 |
| 13 | 4 → 4, 2, 0 | 0 → 0 | (positive 4, negative 0, nonreal 0); (positive 2, negative 0, nonreal 2); (positive 0, negative 0, nonreal 4) |
| 14 | 3 → 3, 1 | 2 → 2, 0 | (positive 3, negative 2, nonreal 0); (positive 3, negative 0, nonreal 2); (positive 1, negative 2, nonreal 2); (positive 1, negative 0, nonreal 4) |
| 15 | 4 → 4, 2, 0 | 2 → 2, 0 | (positive 4, negative 2, nonreal 0); (positive 4, negative 0, nonreal 2); (positive 2, negative 2, nonreal 2); (positive 2, negative 0, nonreal 4); (positive 0, negative 2, nonreal 4); (positive 0, negative 0, nonreal 6) |
| 16 | 4 → 4, 2, 0 | 1 → 1 | (positive 4, negative 1, nonreal 0); (positive 2, negative 1, nonreal 2); (positive 0, negative 1, nonreal 4) |
| 17 | 5 → 5, 3, 1 | 2 → 2, 0 | (positive 5, negative 2, nonreal 0); (positive 5, negative 0, nonreal 2); (positive 3, negative 2, nonreal 2); (positive 3, negative 0, nonreal 4); (positive 1, negative 2, nonreal 4); (positive 1, negative 0, nonreal 6) |
| 18 | 4 → 4, 2, 0 | 1 → 1 | Factor x² first; (positive 4, negative 1, nonreal 0, zero 2); (positive 2, negative 1, nonreal 2, zero 2); (positive 0, negative 1, nonreal 4, zero 2) |
Common Descartes' Rule of Signs Mistakes
Counting zero coefficients
Sign changes are counted between consecutive nonzero coefficients. A missing term does not create a sign.
Changing every sign in f(-x)
Only odd-power terms change sign when x is replaced by -x.
Forgetting the “minus 2” rule
Three sign changes means 3 or 1 roots, not automatically exactly 3.
Calling all remaining roots imaginary
The remaining roots are nonreal complex zeros. For real coefficients, they occur in conjugate pairs.
Ignoring multiplicity
Descartes' Rule counts zeros with multiplicity.
Counting x = 0 as positive or negative
Factor out x first. Zero is neither positive nor negative.
Related Polynomial Worksheets and References
Additional free education references
Created and reviewed by Berke Sahbazoglu
Owner and lead tutor, Burke Tutoring in Fremont. 10+ years of tutoring experience and 6,000+ tutoring hours across K-12 math and science, including extensive Algebra I & II instruction. These problems were adapted from polynomial-zero practice I use with students and independently checked before publication. This resource was reviewed for sign-change counts, f(-x), multiplicity, zero-root handling, and positive / negative / nonreal root profiles.