Upper and Lower Bound Tests

Synthetic division checks that certify no real zero lies beyond a given number.

The Upper and Lower Bound Tests use the bottom row of a synthetic division by $x - c$ to certify that no real zero of $f$ exceeds $c$ (upper bound, all-nonnegative row) or falls below $c$ (lower bound, alternating-sign row), letting you stop testing candidates beyond that point.

Examples

A bottom row of $1, 2, 3, 18$ from dividing by $x - 4$ confirms $4$ as an upper bound.