Math
Descartes' Rule of Signs Calculator
Type a polynomial and the Descartes' rule of signs calculator writes out the sign pattern of f(x), counts the changes, builds f(-x) and counts again, then lists every possible combination of positive, negative, and nonreal (complex) zeros that fits the degree.
Any degree up to 12. Missing powers are fine; zero coefficients are skipped when counting signs.
Try an example
Result
Possible positive real zeros
—
- Possible negative real zeros
- —
- Signs of f(x)
- —
- Sign changes in f(x)
- —
- f(-x)
- —
- Signs of f(-x)
- —
- Sign changes in f(-x)
- —
More details (4 more)
- Is 0 a root?
- —
- Degree (total zeros)
- —
- Possible combinations
- —
- Minimum nonreal zeros
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What Descartes' rule of signs tells you
Write the polynomial in descending order and look at the signs of the nonzero The number multiplying a variable, like the 3 in 3x.. The number of positive real zeros equals the number of sign changes or is less than it by an even number (2, 4, ...). Replace x with -x and repeat to get the possible numbers of negative real zeros. Whatever is left over, up to the degree, must be nonreal complex zeros, which always come in pairs.
Example: f(x) = x^3 - 3x^2 + 2x - 1
| Polynomial | Signs | Changes | Possible zeros |
|---|---|---|---|
| f(x) = x^3 - 3x^2 + 2x - 1 | + − + − | 3 | 3 or 1 positive |
| f(-x) = -x^3 - 3x^2 - 2x - 1 | − − − − | 0 | 0 negative |
So the cubic has either 3 positive real zeros, or 1 positive real zero and 2 nonreal zeros.
Common mistakes
- Counting a zero coefficient as a sign. Skip missing terms entirely.
- Flipping the sign of every term for f(-x). Only odd powers change sign.
- Forgetting the 'less by an even number' part: 3 sign changes means 3 or 1 positive zeros, not exactly 3.
- Ignoring x = 0, which is neither positive nor negative. Factor out x first.
How to use it
- Type f(x) in any order; the calculator sorts it by descending degree.
- Read the possible counts of positive and negative real zeros at the top.
- Use the combinations list to see how many zeros must be nonreal in each case.
How to read the answer
Each combination is a scenario the polynomial could satisfy. The degree equals positive + negative + nonreal zeros (plus any zero roots). Nonreal zeros come in conjugate pairs, so their count is always even.
Common mistakes and edge cases
- Reporting the number of sign changes as the definite number of zeros.
- Miscounting f(-x) when the polynomial has only even powers (then f(-x) = f(x)).
Worked examples
Cubic with 3 sign changes
x^3 - 3x^2 + 2x - 1 → 3 or 1 positive, 0 negative
Possible positive real zeros
3 or 1
Quartic
2x^4 - x^3 + 3x - 5 → 3 or 1 positive; 1 negative
Possible positive real zeros
3 or 1
No positive zeros
x^3 + 2x^2 + x + 5 has no sign changes
Possible positive real zeros
0
Even powers only
x^4 - 5x^2 + 4 → 2 or 0 positive and 2 or 0 negative
Possible positive real zeros
2 or 0
Zero root factored out
x^3 - x = x(x^2 - 1) → 1 positive, 1 negative, plus x = 0
Possible positive real zeros
1
Constant only
A constant has no zeros to count
Possible positive real zeros
Error
Frequently asked questions
What does Descartes' rule of signs say?+
The number of positive real zeros of a polynomial equals the number of sign changes between consecutive nonzero coefficients, or is less than that by an even number. Applying the same count to f(-x) gives the negative real zeros.
Why 'less by an even number'?+
Nonreal zeros of a real polynomial come in conjugate pairs. Each pair removes two real zeros from the count, so the true number of positive zeros differs from the sign-change count by 0, 2, 4, and so on.
How do I find f(-x)?+
Replace every x with -x. Even powers stay the same because (-x)^2 = x^2; odd powers flip sign because (-x)^3 = -x^3. So change the sign of every odd-degree term.
What about x = 0?+
Zero is neither positive nor negative, so factor out the highest power of x first. The calculator does this and reports the multiplicity of the zero root separately.
Can the rule tell me the exact number of zeros?+
Only when there is a single possibility, for example 1 sign change means exactly 1 positive real zero. Otherwise it gives a list of possibilities; use the rational zeros calculator or a graph to narrow it down.
Do I count zero coefficients?+
No. Skip missing terms and compare only the signs of the nonzero coefficients.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Rational Zeros Calculator
Apply the rational root theorem: list every candidate ±p/q for a polynomial, test each one exactly, report the actual rational zeros with multiplicity, and show the deflated (remaining) factor.
Cubic Equation Calculator
Solve ax^3 + bx^2 + cx + d = 0 exactly when a rational root exists (rational root theorem + synthetic division + quadratic formula), otherwise with Cardano or the trigonometric method, with discriminant, factored form, complex roots, and a graph.
Polynomial Division Calculator
Divide polynomials by long division or synthetic division. Type both polynomials and get the quotient, remainder, the full synthetic-division tableau or long-division rows, the check, and every step.
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with exact simplified radicals, reduced fractions, or complex roots, plus discriminant, vertex, factored and vertex forms, a graph, and every step.
Discriminant Calculator
Find the discriminant b^2 - 4ac of a quadratic, classify the roots as rational, irrational, repeated, or complex, and preview the exact roots with steps.
Completing the Square Calculator
Complete the square on ax^2 + bx + c step by step: every algebraic line shown, exact fractions, vertex form a(x - h)^2 + k, the vertex, and the roots as simplified radicals.
Last updated: September 4, 2026