Descartes Rule of Signs - Steps & Examples
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Descartes Rule of Signs - Steps & Examples
TL;DR
Descartes rule of signs counts the sign changes in a polynomial's coefficients to bound how many positive and negative real roots it can have — the count of positive roots equals the number of sign changes in f(x), or less by an even number. This article gives the steps, a possible-roots chart, the link to complex roots, six worked examples, and the mistakes that throw off the count.
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Bhanzu Team Last updated on June 26, 2026 12 min read
What is Descartes Rule of Signs?
Descartes rule of signs states that the number of positive real roots of a polynomial f(x) is either equal to the number of sign changes between consecutive nonzero coefficients, or less than that by an even number. Applying the same count to f(−x) gives the possible number of negative real roots.
Written out, the rule has two parts:
- Positive roots: count the sign changes in f(x) as you read the coefficients in order. The number of positive real roots is that count, or that count minus 2, minus 4, and so on, down to 0 or 1.
- Negative roots: form f(−x), count its sign changes, and the number of negative real roots is that count or less by an even number.
The "less by an even number" wording is there because real roots that are not present show up as complex conjugate pairs, and each pair removes two from the count.
Variable Glossary:
| Symbol | Meaning |
|---|---|
| f(x) | the polynomial, written in descending powers |
| f(−x) | the polynomial with x replaced by −x, used for negative roots |
| sign change | a place where consecutive nonzero coefficients have opposite signs |
| degree | the total number of roots, counting complex roots and multiplicity |
How Do You Apply Descartes Rule of Signs?
The procedure is mechanical once you write the polynomial in standard form. Take f(x)=x³−3x−2:
- Write f(x) in descending powers and list the nonzero coefficient signs. Here: +,−,− (the x² term is missing, so skip it).
- Count sign changes in f(x). From + to − is one change; from − to − is none. Total: 1 sign change, so there is exactly 1 positive root.
- Form f(−x): (−x)³−3(−x)−2=−x³+3x−2. Signs: −,+,−.
- Count sign changes in f(−x). From − to + is one; from + to − is one. Total: 2 changes, so there are 2 or 0 negative roots.
- Account for complex roots. The degree is 3, so total roots = 3. The possible splits are recorded in a chart.
The possible-roots chart
| Positive roots | Negative roots | Complex roots | Total |
|---|---|---|---|
| 1 | 2 | 0 | 3 |
| 1 | 0 | 2 | 3 |
Each row must sum to the degree, 3.
Examples of Descartes Rule of Signs
Example 1
Find the possible number of positive real roots of f(x)=x⁴−3x²+2x−1.
Signs of the nonzero coefficients: +,−,+,−. Count the changes: +→−, −→+ — three sign changes.
So the number of positive real roots is 3 or 1.
Final answer: 3 or 1 positive real roots.
Example 2
Find the possible number of negative real roots of f(x)=x³+2x²+5x+4.
Correct. Form f(−x):
f(−x)=−x³+2x²−5x+4.
Signs: −,+,−,+. Count changes: −→+ (twice) — three changes.
So the number of negative real roots is 3 or 1.
Final answer: 3 or 1 negative real roots.
Example 3
Determine the possible number of positive and negative roots of f(x)=x⁵+4x⁴−3x²+x−6.
Positive: signs are +,+,−,+,−. Changes: +→− (three changes). Negative: f(−x)=−x⁵+4x⁴−3x²−x−6.
Final answer: positive roots 3 or 1; negative roots 2 or 0.
Example 4
Find the exact number of positive roots of f(x)=x²+x+1.
Signs: +, +, + — zero sign changes, so there are exactly 0 positive roots.
Final answer: 0 positive roots.
Example 5
Build the complete roots chart for f(x)=x⁴−1.
Positive: signs +, −, +. One change, so exactly 1 positive root.
Negative: f(−x)=x⁴−1 , same signs +, −, + — one change, so 1 negative root.
Final answer: 1 positive, 1 negative, 2 complex roots.
Example 6
Use the rule to bound the complex roots of f(x)=x³−x²+x−1.
Positive: signs +, −, +, − — three changes, so 3 or 1 positive roots. Negative: f(−x)=−x³−x²−x−1, zero changes, so 0 negative roots.
Final answer: either 3 positive and 0 complex, or 1 positive and 2 complex.
The Mathematicians Behind Descartes Rule of Signs
René Descartes (1596–1650, France) published the rule in his 1637 work La Géométrie, which founded analytic geometry.
Carl Friedrich Gauss (1777–1855, Germany) later sharpened the rule by proving it always counts less by an even number.
Why This Rule Earns Its Keep
Descartes rule of signs helps in narrowing the root search before computation. On its own it does not find roots, but limits the candidates.
Where Students Trip Up On Descartes Rule of Signs
Mistake 1: Using f(x) to count negative roots
Don’t do this: Concluding f(x) has zero negative roots because there are no sign changes.
Mistake 2: Counting zero-coefficient terms as sign changes
The correct way: Only nonzero coefficients count; skip terms when reading sign changes.
Mistake 3: Treating the count as exact instead of a maximum
The sign-change count is a maximum.
Practice Questions on Descartes Rule of Signs
- Find the possible number of positive real roots of f(x)=x³+2x²+x+5.
- Find the possible number of negative real roots of the same f(x).
- Find the possible number of positive real roots of f(x)=x⁴−x³+x²−x+1.
- Build the full roots chart for f(x)=x³−1.
- State the possible positive and negative roots of f(x)=2x⁴−5x³−x+6.
- How many positive roots does f(x)=x²+3x+5 have?
Answers
- 0 positive roots.
- 3 or 1 negative roots.
- 4, 2, or 0 positive roots.
- 1 positive, 0 negative, 2 complex roots.
- 2 or 0 positive; 0 negative roots.
- 0 positive roots (and both roots are a complex conjugate pair).
Key Takeaways
- Descartes rule of signs bounds the number of positive and negative real roots.
- Positive roots come from f(x); negative roots come from f(−x) — always form f(−x) separately.
- The sign-change count is a maximum; true numbers are counts or less by two.
- Complex roots fill the gap between the maximum real roots and the degree, and always show as conjugate pairs.
A Practical Next Step
Work through these practice questions to reinforce your understanding.