Synthetic Division Calculator
Divide polynomials using synthetic division and find quotient and remainder.
Divide P(x) by (x − c). Enter coefficients from highest to lowest degree (include 0 for missing terms).
Include zeros for missing degree terms: x³ + 2 → "1 0 0 2"
Quick examples:
What Is the Synthetic Division Calculator?
The Synthetic Division Calculator divides any polynomial P(x) by a linear factor (x − c) using the compact synthetic division algorithm. Enter the coefficients from highest to lowest degree (include zeros for missing terms) and the value c. The calculator shows the full synthetic division tableau with the carries row, the quotient polynomial Q(x), the remainder R, and confirms whether c is a root via the Factor and Remainder Theorems.
- ›Enter coefficients space-separated: x³−6x²+11x−6 → "1 -6 11 -6"
- ›Include 0 for missing degree terms: x³+2 → "1 0 0 2"
- ›Remainder = 0 confirms c is a root and (x−c) is a factor
- ›Quotient is one degree lower than the dividend
- ›The Remainder Theorem states R = P(c), fast polynomial evaluation
Formula
Synthetic Division Algorithm
Dividend
P(x) = aₙxⁿ + ... + a₁x + a₀
Divisor
(x − c); c is the root/test value
Algorithm
bring down → ×c → add → repeat
Remainder Thm
P(c) = remainder
Factor Thm
P(c)=0 ↔ (x−c) is a factor
Result
P(x) = (x−c)Q(x) + R
How to Use
- 1Enter the polynomial coefficients (space or comma separated), highest degree first
- 2Include zeros for any missing degree terms
- 3Enter the value c for the divisor (x − c)
- 4Click Divide
- 5The synthetic tableau shows the carries and result row with the quotient and remainder
- 6Use Quick Examples to load preset polynomials like x³−6x²+11x−6 ÷ (x−2)
Example Calculation
Divide x³ − 6x² + 11x − 6 by (x − 2):
2 | 1 −6 11 −6
| 2 −8 6
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
1 −4 3 0 ← R = 0
Q(x) = x² − 4x + 3, Remainder = 0
P(x) = (x−2)(x²−4x+3) = (x−2)(x−1)(x−3)
Evaluate P(3) = 3³ − 2·3² + 1 for P(x) = x³−2x²+1:
3 | 1 −2 0 1
| 3 3 9
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
1 1 3 10 ← R = P(3) = 10
Horner's Method
Synthetic division is identical to Horner's method for polynomial evaluation. Instead of computing each power of x separately (aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ...), Horner's method nests the computation: (...((aₙ·x + aₙ₋₁)·x + aₙ₋₂)·x + ...). This requires only n multiplications and n additions, far more efficient, and numerically stable. It is the standard algorithm in computer polynomial evaluation.
Understanding Synthetic Division
Synthetic Division Step-by-Step Reference
| Example | Polynomial | c | Quotient | Remainder | Root? |
|---|---|---|---|---|---|
| 1 | x³−6x²+11x−6 | 1 | x²−5x+6 | 0 | Yes |
| 2 | x³−6x²+11x−6 | 2 | x²−4x+3 | 0 | Yes |
| 3 | x³−6x²+11x−6 | 3 | x²−3x+2 | 0 | Yes |
| 4 | x³−6x²+11x−6 | 4 | x²−2x+3 | 6 | No |
| 5 | x⁴−1 | 1 | x³+x²+x+1 | 0 | Yes |
| 6 | x³+1 | −1 | x²−x+1 | 0 | Yes |
| 7 | 2x²+x−6 | 1.5 | 2x+4 | 0 | Yes |
| 8 | x³−2x²+1 | 1 | x²−x−1 | 0 | Yes (x=1) |
Frequently Asked Questions
When can I use synthetic division?
- ›Works for: (x−2), (x+3) = (x−(−3)), (x−0.5)
- ›Does NOT work directly for: (x²−1), (2x−3) as-is, (x²+x+1)
- ›For (2x−3): find c = 3/2 = 1.5, divide by (x−1.5), divide quotient by 2
- ›For higher-degree divisors: use polynomial long division
What does the Remainder Theorem say?
The Remainder Theorem makes synthetic division doubly useful: it both divides the polynomial AND evaluates it at a point in the same pass.
- ›P(x) = (x−c)·Q(x) + R, and R = P(c)
- ›To find P(2): synthetic divide by (x−2); remainder is P(2)
- ›Faster than direct substitution for high-degree polynomials
- ›Basis of Horner's method for efficient polynomial evaluation
What does the Factor Theorem say?
- ›P(c)=0 ↔ (x−c) is a factor of P(x)
- ›After confirming a factor, Q(x) is the deflated polynomial
- ›Factor x³−6x²+11x−6 completely: test x=1,2,3 → all roots → (x−1)(x−2)(x−3)
- ›Rational Root Theorem gives candidates for c: factors of a₀ / factors of aₙ
What is the Rational Root Theorem?
For P(x) = 2x³ − 3x² − 11x + 6, candidates are ±{1,2,3,6}/±{1,2} = ±1, ±2, ±3, ±6, ±1/2, ±3/2. Testing c=3 gives remainder 0 → (x−3) is a factor.
- ›Candidates: ±(factors of a₀) / ±(factors of aₙ)
- ›Test each candidate via synthetic division
- ›Once a root c is found, divide by (x−c) and repeat on Q(x)
- ›Not all polynomials have rational roots (e.g., x²−2 has irrational roots ±√2)
How do I handle missing degree terms?
- ›x³ + 2 → coefficients: 1 0 0 2
- ›x⁴ − 1 → coefficients: 1 0 0 0 -1
- ›2x³ + x → coefficients: 2 0 1 0
- ›Count: degree n polynomial needs exactly n+1 coefficients
What is the difference between synthetic and long division?
- ›Synthetic: only coefficients, 3 rows, works for (x−c) divisors only
- ›Long division: writes out full polynomial expressions, any degree divisor
- ›Both: same quotient Q(x) and remainder R
- ›Synthetic is ~3× faster to compute by hand for linear divisors
Is this synthetic division calculator free?
Yes, completely free with no registration required. All calculations run locally in your browser.