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Synthetic Division Calculator

Math Calculators

Synthetic & long division · remainder theorem · rational roots

Quick Answer

Synthetic Division Calculator: Synthetic & long division · remainder theorem · rational roots. Synthetic Division Calculator uses standard math formulas to compute accurate results instantly.

Quotient
3x² + 6x + 10
Remainder 25 · P(2) = 25 · (x − 2) is not a factor
Degree3
Remainder25
P(2)25
Factor?No

Synthetic division tableau

230−25
×c61220
+361025

Row 1: coefficients · Row 2: c×previous result · Row 3: sums (last cell = remainder)

Calculation steps

  1. 1Polynomial P(x) = 3x³ − 2x + 5, coefficients high→low: [3, 0, −2, 5] (missing terms padded with 0)
  2. 2Synthetic division with c = 2: bring down the leading coefficient, then repeat bᵢ = aᵢ + c·bᵢ₋₁
  3. 3Result row gives the quotient coefficients and the final number is the remainder: quotient = 3x² + 6x + 10, remainder = 25
  4. 4By the remainder theorem, P(2) = 25 — same as direct substitution
  5. 5Remainder is not 0, so (x − 2) is not a factor

Key theorems

TheoremStatementUseExample
Remainder theoremRemainder of P(x)÷(x−c) equals P(c)Fast evaluation & checkingP(2)=25 ⟺ remainder 25
Factor theoremRemainder 0 ⟺ (x−c) is a factorDivisibility & root findingx=1 rem 0 → (x−1) factor
Rational root theoremAny rational root is ±p/q with p|constant, q|leading coefNarrows candidates to a finite set2x³+… → ±1, ±2, ±1/2
Horner's ruleSame recurrence as synthetic divisionFastest evaluation by hand or codeOnly n multiplies & n adds
DeflationEach found root lowers the degree by oneSolve all roots step by stepCubic → quadratic → formula
Pad missing termsCoefficients run high→low with zeros for gapsInput conventionx³+5 → enter 1 0 0 5

Coefficients are entered from the highest degree down, separated by spaces or commas — pad missing terms with 0 (x³+5 → 1 0 0 5).

Synthetic division only works for a linear divisor (x−c); for a general divisor use long division.

The rational root theorem only finds rational roots; irrational or complex roots appear in the final quadratic (or higher) factor.

Horner's rule and synthetic division are the same recurrence — the fastest way to evaluate a polynomial by hand.

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How to Use

  1. Enter values or expressions: type directly into the Synthetic Division Calculator — keyboard friendly.
  2. Instant results: everything computes live as you type, no button needed.
  3. Follow the steps: the result panel shows full working and formula substitution.
  4. Copy or recompute: one-click copy; change any input to recalculate instantly.

Key Points

  • Synthetic Division Calculator is a free online math tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: Synthetic & long division · remainder theorem · rational roots.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Synthetic Division Calculator formula

Synthetic Division Calculator uses standard math formulas to compute accurate results instantly.

FAQ

How do I use the Synthetic Division Calculator?
Open the page and fill in the input fields — results update automatically. Synthetic & long division · remainder theorem · rational roots.
Are the results accurate?
Yes. All calculations use standard mathematical formulas and are verified for accuracy.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required — just open and use.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. All calculations run locally in your browser. Nothing is uploaded or stored.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated:

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Results provided by this tool are for reference only and do not constitute professional advice.