Count signs, not roots
Supply one coefficient and its matching power for each term. Add or remove entries in both lists to change the number of terms. Repeated powers are combined and terms are sorted before counting. Zero coefficients are ignored. Expression mode supports polynomial arithmetic and parentheses, up to degree 64.
Descartes’ rule says the positive-root count equals the sign changes in P(x) minus an even nonnegative integer. Apply the same rule to P(−x) for negative roots. Counts include multiplicities. Zero roots are counted separately; remaining nonreal roots occur in conjugate pairs. The table lists candidates, not proof that every candidate occurs for this polynomial. See OpenStax.
Why your number may differ
We remove zero terms, combine repeated powers and never list negative counts. The reference can show −1 as a possible negative-root count at the defaults; root counts cannot be negative. A zero polynomial has infinitely many roots and is rejected rather than assigned a finite count.