Dividing the polynomial x³ − 2x² + 5x − 1 by (x − 3) the long way is tedious — but the Remainder Theorem hands you the remainder in one substitution: evaluate the polynomial at x = 3. Computing 27 − 18 + 15 − 1 gives 23, so the remainder is 23, no long division required. The theorem states that dividing P(x) by (x − a) leaves remainder P(a), which turns every polynomial division remainder into a simple evaluation.

Its famous corollary is the Factor Theorem: if P(a) = 0, then (x − a) is a factor — the engine behind root-finding, from classroom factoring to the rational root theorem to computer algebra systems. Engineers use it to test candidate roots of characteristic equations without full division. This calculator multiplies out one evaluation term at a time to help verify your substitution: for P(3) above, type the x³ coefficient's contribution — 1 × 27 = 27, so enter 27 — in the Variable A box, and type 1 in the Variable B box. The Result box passes 27 through; repeat for each term (coefficient × power of a), then add the terms on paper: 27 − 18 + 15 − 1 = 23, the remainder. If the total is 0, you have found a factor.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Remainder Theorem Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.