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Descartes' Rule of Signs Calculator

Sign changes in f(x) and f(−x) marked one by one, then every possible (positive, negative, complex) combination that adds up to the degree.

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Descartes' Rule of Signs Calculator is free to use as often as you like, directly from this page. It covers positive, negative, complex — adjust any of them and the result follows immediately.

How to use it

  1. Open the tool — no signup or install needed.
  2. Enter your input or adjust the available options.
  3. Get your result instantly, then copy or download it.

Frequently asked questions

What does Descartes' Rule of Signs Calculator do?

Sign changes in f(x) and f(−x) marked one by one, then every possible (positive, negative, complex) combination that adds up to the degree.

What does it take into account?

It factors in positive, negative, complex. Change any of them and the output follows immediately.

When would I actually use this?

Getting through a problem set: solving for the unknown, factoring an expression, and — more usefully — seeing the steps that got there.

What is the most common mistake?

Losing a solution when both sides are squared or divided by an expression. Squaring can add roots that do not satisfy the original equation, and dividing can remove one — check every answer back in the original.

How is Descartes' Rule of Signs Calculator different from Cramer's Rule Calculator?

They sit next to each other but answer different questions: Cramer's Rule Calculator is the one to open when you need it to solves a 2×2 or 3×3 system with exact fractions, shows every determinant expanded term by term, and when D = 0 separates the two cases most calculators merge: dependent with infinitely many solutions, or inconsistent with none. Pick whichever matches what you're starting from — both are free.

Is there a tool for the next step?

Simpson's Rule Calculator is the closest one after this: Integrate with the 1/3 and 3/8 rules, with the weight pattern shown, the subinterval count adjusted when it must be, and the error bound.

What else is worth having open alongside it?

Trapezoidal Rule Calculator and Rule of three — they come up in the same task often enough to be worth a second tab.

Where do the figures come from?

The algebra is symbolic where the expression allows it and numeric otherwise, and the tool says which it used. Exact results are given as fractions or radicals rather than rounded decimals.

Further reading

All guides →
How-toHow to Use the Quadratic Formula: Solving ax² + bx + c = 0Solve any quadratic equation with x = (−b ± √(b² − 4ac)) / 2a, and read the discriminant to know how many real solutions to expect.How-toHow to Find a Cube Root by HandA step-by-step guide to finding a cube root by estimation and refinement — plus how it differs from a square root and when a value stays irrational.ExplainerHow Exponents Work: Powers and the Laws of ExponentsAn exponent is repeated multiplication. Learn the laws of exponents, what negative and fractional powers mean, and why anything to the zero is one.ExplainerThe Flat Pattern of a Cone Is a Sector, Not a CircleRoll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.ExplainerA 1 % Drop Rate Does Not Mean One Hundred RunsAt 1 %, a hundred attempts give you 63.4 % — not certainty. Ninety per cent takes 230 attempts and ninety-nine takes 459, and more than a third of players are still empty-handed at a hundred.ExplainerFibonacci and the Golden RatioThe Fibonacci sequence adds each pair of terms to make the next; the ratio of neighboring terms homes in on the golden ratio φ ≈ 1.618. See how, and where the pattern turns up.