Determine the possible number of positive and negative real roots of any polynomial. Enter your polynomial as text or by coefficients — get a color-coded sign sequence, full step-by-step analysis, and a complete possibilities table.
| Positive Real | Negative Real | Non-real Complex Pairs |
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Sign change: consecutive non-zero coefficients have opposite signs
Descartes' Rule looks at the non-zero coefficients of a polynomial in descending order and counts how many times the sign switches from + to − or from − to +. Zero coefficients are skipped.
The number of positive real roots equals the number of sign changes in f(x), or that number minus any even number (2, 4, 6, …). This is because complex roots always come in conjugate pairs — losing two real roots means gaining one complex conjugate pair.
For negative real roots, substitute x → −x first. The sign of the term with xn changes by a factor of (−1)n — even powers are unchanged, odd powers flip. Then count sign changes in f(−x) the same way.
Total roots = degree (counting multiplicity, complex included)
Step 1: Count sign changes in f(x) → possibilities for positive real roots (subtract 2 at a time).
Step 2: Count sign changes in f(−x) → possibilities for negative real roots (subtract 2 at a time).
Step 3: Pair every combination. The non-real complex roots always come in conjugate pairs: if pos + neg = k, then complex pairs = (degree − k)/2.
Only keep combinations where complex pairs ≥ 0 and is a whole number. The table in the results shows every valid combination.
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