How do I use the Descartes' Rule of Signs Calculator?

Enter the coefficients of your polynomial in descending order of degree (highest degree first), including zeros for missing terms. The calculator will count sign changes and show the possible number of positive and negative real roots.

What does 'sign changes' mean in this context?

A sign change occurs when consecutive non-zero coefficients have opposite signs (positive to negative or negative to positive). The number of sign changes in the polynomial gives the maximum possible number of positive real roots.

How are negative roots determined?

To find possible negative roots, the calculator evaluates the polynomial at -x, which means it alternates the signs of coefficients for odd-degree terms. Then it counts sign changes in that new polynomial to give the maximum possible number of negative real roots.

数学

即时、私密且免费

Descartes' Rule of Signs Calculator.

Count sign changes in polynomial coefficients and the possible positive and negative roots.

设备端计算无需注册

输入数值

输入时结果会更新。

02 / 答案

Coefficients (highest degree first)

结果

signChanges: 2.000000 · possiblePositiveRoots: 2.000000 · possibleNegativeRoots: 1.000000

Use Cases

Quickly analyze polynomial roots

Determine the possible number of positive and negative real roots for any polynomial, which helps in graphing or further root-finding.

Example: For x^3 - 2x^2 - x + 2, enter coefficients 1, -2, -1, 2 to see possible positive roots (2 or 0) and negative roots (1).

Check polynomial behavior before solving

Use the rule to narrow down the number of real roots before applying numerical methods or factoring.

Example: For a quintic, quickly see if it can have 3 positive roots and 2 negative roots, guiding your solving strategy.

Frequently Asked Questions

How do I use the Descartes' Rule of Signs Calculator?: Enter the coefficients of your polynomial in descending order of degree (highest degree first), including zeros for missing terms. The calculator will count sign changes and show the possible number of positive and negative real roots.
What does 'sign changes' mean in this context?: A sign change occurs when consecutive non-zero coefficients have opposite signs (positive to negative or negative to positive). The number of sign changes in the polynomial gives the maximum possible number of positive real roots.
How are negative roots determined?: To find possible negative roots, the calculator evaluates the polynomial at -x, which means it alternates the signs of coefficients for odd-degree terms. Then it counts sign changes in that new polynomial to give the maximum possible number of negative real roots.

Tips & Common Mistakes

Tips

Ensure you include zero coefficients for any missing terms (e.g., for x^3 - 1, enter 1, 0, 0, -1).
The calculator counts sign changes only between consecutive non-zero coefficients; zeros are ignored.
The possible number of positive roots is the number of sign changes minus an even integer (e.g., 3, 1, or none).
For negative roots, the calculator automatically substitutes -x, so you don't need to manually alter the polynomial.

Common Mistakes to Avoid

Forgetting to include zero coefficients for missing terms, which can lead to incorrect sign change counts.
Misinterpreting the result: the calculator gives possible numbers, not the exact count of real roots.
Entering coefficients in ascending order (constant first) instead of descending order (highest degree first).

Last updated: August 13, 2026