A polynomial like p(x) = x³ − 6x² + 11x − 6 is a machine: feed it an x, and it cranks through the arithmetic to produce a y. Evaluating at x = 2 gives 8 − 24 + 22 − 6 = 0 — so x = 2 is a root, a place where the graph crosses the axis. That evaluate-and-check loop is the heartbeat of algebra: every root hunt, every graph sketch, every "find p(3)" homework question is just running the machine and reading the dial. Engineers evaluating a cost model at different production levels and game developers sampling a curve are doing the same thing at scale.

The derivatives are the machine's speedometer and accelerometer. p′(x) = 3x² − 12x + 11 tells you the slope — where the graph climbs or falls — and its zeros mark the local peaks and valleys. p″(x) = 6x − 12 tells you the concavity, where the curve bends upward or downward, and its zero marks the inflection point. This calculator evaluates all three at your chosen x: enter the coefficients a, b, c, d and your x-value, then read p(x), p′(x), and p″(x). For the example at x = 2: p = 0 (a root), p′ = −1 (gently falling through the axis), p″ = 0 (an inflection point) — a complete portrait of the graph at that single point.

With the pre-filled defaults — a (x³ coefficient) 1, b (x² coefficient) -6, c (x coefficient) 11, d (constant term) -6, Evaluate at x = 2 — the Polynomial Calculator returns p(x) 0.000000, p′(x) at x -1.000000, p″(x) at x 0.000000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.