What does this calculator do?

The calculator counts sign changes in f(x) and f(−x) and reports possible counts of positive and negative real roots.

It does not solve the roots themselves. It is a counting and narrowing tool, often used before rational-zero testing or before solving a polynomial by another method.

How sign changes are counted

A sign change occurs when adjacent nonzero terms switch from positive to negative or from negative to positive.

• The sequence +, −, + has two sign changes.
• The sequence +, +, − has one sign change.
• Zero-coefficient gaps are skipped.
• Terms are read in descending degree order.

Why f(−x) is used

Descartes' rule is applied to f(−x) to estimate the possible number of negative real roots.

When f(−x) is formed, odd-degree terms change sign and even-degree terms keep their sign. The calculator builds this transformed polynomial automatically.

How to read possible root counts

The rule gives possible counts, not exact root values. If f(x) has 3 sign changes, the possible positive real root counts are 3 or 1.

If f(−x) has 2 sign changes, the possible negative real root counts are 2 or 0.

The calculator also shows a derived non-real root-pair count based on the degree and the maximum possible real-root counts.

Worked example

For f(x)=x^4−3x^3+2x^2−x+5, the sign sequence is +, −, +, −, +.

1. + to − gives the first sign change.
2. − to + gives the second sign change.
3. + to − gives the third sign change.
4. − to + gives the fourth sign change.

So the possible positive real root counts are 4, 2, or 0. A separate root-finding method is needed to know which case actually occurs.

Using it with the Rational Zeros Theorem

Descartes' Rule of Signs is useful before testing rational-zero candidates because it can narrow expectations about sign direction.

• If possible positive root count is 0, positive rational zeros should not be expected.
• If the negative count is small, negative candidates can be interpreted with that limit in mind.
• The rule does not generate candidates; it only limits counts.

That makes the rational zeros calculator a natural next step.

What this calculator does not solve

This calculator does not return actual root values. It will not tell you that x=2 or x=−1 is a root.

• Use a rational zeros calculator to test rational candidates.
• Use the quadratic formula for second-degree equations.
• Use a cubic equation calculator for cubic equations.
• The zero polynomial is rejected because the rule is undefined for it.

Common mistakes

• Counting missing zero terms as sign changes.
• Forgetting to change signs of odd-degree terms in f(−x).
• Treating possible positive roots as confirmed roots.
• Confusing non-real roots with negative real roots.
• Using the rule as a complete root solver.