Descartes' Rule of Signs Calculator

Estimate the possible number of positive and negative real roots of a cubic.

Max positive real roots 2

Max negative real roots 1

Formula: Count sign changes in f(x) and f(−x)

Step-by-step with your numbers:

1. Values used:

2. a (x³) = 1

3. b (x²) = -3

4. c (x) = -4

5. d (constant) = 12

7. Max positive real roots = 2

8. Max negative real roots = 1

Did we solve your problem today?

Descartes' Rule of Signs bounds how many positive and negative real roots a polynomial can have.

The math behind it

The number of positive real roots equals the sign changes in f(x), or less by an even number. Negative roots use the sign changes in f(−x).

x³ − 3x² − 4x + 12 has 2 sign changes → up to 2 positive roots.

Does it give the exact count?

No — it gives the maximum; the actual number can be lower by a multiple of 2 (accounting for complex pairs).