Descartes' Rule of Signs Calculator
Determine the possible number of positive, negative, and non-real roots of any polynomial
Enter Polynomial Coefficients
Enter coefficients from lowest degree (a₀) to highest degree. Fields will appear as you type.
Getting Started
Enter polynomial coefficients starting from the constant term (a₀) to analyze the possible number of roots using Descartes' rule of signs.
Example Calculations
Example 1: p(x) = 6x⁵ + 5x⁴ - 4x³ + 3x² + 2x + 1
Coefficients: [1, 2, 3, -4, 5, 6]
Sign changes in p(x): 2 (from +5 to -4, from -4 to +3)
Possible positive roots: 2 or 0
Sign changes in p(-x): 3
Possible negative roots: 3 or 1
Minimum non-real roots: 5 - 2 - 3 = 0
Example 2: p(x) = x³ - 2x² - x
Coefficients: [0, -1, -2, 1]
Zero multiplicity: 1 (x is a factor)
Sign changes in p(x): 1
Possible positive roots: 1
Possible negative roots: 1
Total roots accounted: 1 (zero) + 1 (positive) + 1 (negative) = 3
Example 3: p(x) = x³ + x² + 1
Coefficients: [1, 0, 1, 1]
Sign changes in p(x): 0
Possible positive roots: 0
Sign changes in p(-x): 1
Possible negative roots: 1
Minimum non-real roots: 3 - 0 - 1 = 2
How the Rule Works
Step 1:
Count sign changes in coefficients of p(x) (ignoring zeros)
Step 2:
Possible positive roots = sign changes minus even numbers down to 0 or 1
Step 3:
For negative roots, repeat with p(-x) by alternating coefficient signs
Step 4:
Non-real roots = degree - zero multiplicity - max positive - max negative
Key Concepts
Sign Changes
Count transitions from + to - or - to + in coefficients
Same Parity
Actual roots differ from sign changes by even numbers
Zero Multiplicity
Count leading zero coefficients for roots at x = 0
Quick Tips
Ignore zero coefficients when counting sign changes
The rule gives upper bounds and possible values
Complex roots always come in conjugate pairs
Use with other methods for complete factorization
Most useful for polynomials of degree 3 and higher
Understanding Descartes' Rule of Signs
What is Descartes' Rule of Signs?
Discovered by René Descartes in "La Géométrie," this rule provides a method for determining the possible number of positive real roots, negative real roots, and non-real roots of a polynomial without actually solving it.
The Rule States:
- •The number of positive real roots is at most the number of sign changes in the coefficients
- •The actual number differs from the sign changes by an even number (same parity)
- •For negative roots, apply the same rule to p(-x)
Mathematical Foundation
For polynomial p(x) = aₙxⁿ + ... + a₁x + a₀:
• Count sign changes in [a₀, a₁, a₂, ..., aₙ]
• This gives maximum positive roots
• For p(-x): alternate signs starting from a₁
• Non-real ≥ degree - zero mult. - max pos. - max neg.
Applications
- •Preliminary analysis before factoring
- •Optimization problems in calculus
- •Control theory and stability analysis
- •Mathematical modeling and equation solving
Important Limitations
What the Rule Tells Us
- • Upper bounds on positive/negative roots
- • Possible values (same parity as sign changes)
- • Minimum number of complex roots
What the Rule Doesn't Tell Us
- • Exact number of roots (usually)
- • Location or values of roots
- • Multiplicities of roots