Polynomial Functions

Synthetic Division & the Remainder Theorem

Using synthetic division and the Remainder Theorem to evaluate and factor polynomials.

Advanced20 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Polynomial Long Division

A Faster Shortcut, and Why It Works

Long division of a polynomial by (x − c) carries a lot of repeated bookkeeping — writing out every power of x again and again, when only the coefficients actually change from step to step. Before reading on, think about what could be safely stripped away from the long division process if only the coefficients mattered.

Definition — Synthetic Division

A streamlined version of polynomial long division for dividing by (x − c), using only the polynomial's coefficients and the value c, without rewriting the variable at every step.

Worked Example — Performing Synthetic Division

Divide x³ + 2x² − 5x + 6 by x − 2 using synthetic division — the same division worked through with long division in the previous lesson, giving a quotient of x² + 4x + 3 with remainder 12. Write c = 2 and the coefficients 1, 2, −5, 6. Bring down the 1. Multiply 1 × 2 = 2, add to the next coefficient: 2 + 2 = 4. Multiply 4 × 2 = 8, add: −5 + 8 = 3. Multiply 3 × 2 = 6, add: 6 + 6 = 12. The bottom row, 1, 4, 3, 12, gives quotient x² + 4x + 3 with remainder 12 — matching the long division result exactly, as it should, since both methods perform the same division.

Tip

Synthetic division only works directly for a divisor of the form (x − c) — for a divisor with a leading coefficient other than 1, like (2x − 3), first rewrite it or fall back to long division.

Definition — The Remainder and Factor Theorems

The Remainder Theorem: dividing a polynomial f(x) by (x − c) leaves a remainder equal to f(c). The Factor Theorem: (x − c) is a factor of f(x) exactly when f(c) = 0.

The Remainder Theorem follows directly from the division identity itself: f(x) = (x − c)·q(x) + r, true for any polynomial division. Substituting x = c makes the first term vanish, since (c − c) = 0: f(c) = 0 · q(c) + r = r. Whatever the remainder r is, it must equal f(c) — meaning synthetic division doubles as a fast way to evaluate a polynomial at a specific value, without direct substitution. The Factor Theorem then follows immediately: if the remainder is 0, there's nothing left over, meaning (x − c) divides evenly — exactly what 'is a factor' means.

Worked Example — Using the Remainder Theorem to Evaluate a Polynomial

Find f(3) for f(x) = 2x³ − 5x² + x − 1 using synthetic division instead of direct substitution. Coefficients: 2, −5, 1, −1, with c = 3. Bring down 2. 2×3=6, −5+6=1. 1×3=3, 1+3=4. 4×3=12, −1+12=11. The remainder is 11, so f(3) = 11 by the Remainder Theorem.

Worked Example — Using the Factor Theorem to Test a Root

Is x = 1 a root of f(x) = x³ − 4x² + x + 6? By the Factor Theorem, check whether f(1) = 0: f(1) = 1 − 4 + 1 + 6 = 4. Since f(1) ≠ 0, x = 1 is not a root, and (x − 1) is not a factor.

Common Mistakes

Using the wrong sign for c in synthetic division, such as using −2 instead of 2 when dividing by (x − 2).
Synthetic division uses the root value c directly — for divisor (x − c), use c itself; for (x − 2), that means using 2, not −2.
Forgetting to include a 0 for any missing-degree coefficient before starting synthetic division.
List every coefficient in order by descending degree, using 0 for any missing term — skipping a missing degree misaligns every subsequent step.

Key Takeaways

Synthetic division streamlines dividing by (x − c) using only coefficients.
The Remainder Theorem says dividing by (x − c) leaves a remainder equal to f(c), following directly from the division identity f(x) = (x − c)q(x) + r.
The Factor Theorem says (x − c) is a factor exactly when f(c) = 0.

Summary

Synthetic division and the Remainder Theorem connect division, evaluation, and factoring into one unified idea. The next unit shifts from whole-number exponents to fractional ones, extending exponent rules to radicals.

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