The Solution
Calculation: Synthetic Division Calculator
Synthetic Division Calculator
Enter your polynomial coefficients above and click "Perform Synthetic Division" to see results.Graph will appear here after you solve.
Cubic Equation Solver
Synthetic Division solver
Find real and complex roots, discriminant, and cubic graphs in seconds with our precision solver.
The Equation
Enter the cubic's coefficients and a divisor value r to perform fast synthetic division by (x − r).
f(x) = 0 = 0
The Treatise
What isSynthetic Division Calculator?
- For a school student: Synthetic division is a shortcut table for dividing polynomials. Long division with x's takes many lines of writing; synthetic division does the same job with just the numbers, arranged in a small grid where you multiply and add your way across. Less writing, same answer.
- For a college student: It is Ruffini's rule for dividing a polynomial by a linear divisor (x − c). Write the coefficients in a row, bring down the first, then repeat: multiply by c, add to the next coefficient. The bottom row gives the quotient coefficients followed by the remainder, which equals P(c) by the remainder theorem.
- For the advanced reader: The tableau is Horner's method for O(n) polynomial evaluation, reorganized as division. Writing P(x) = ((aₙx + aₙ₋₁)x + ...)x + a₀, each multiply-add peels off one level of nesting. This form is also numerically stabler than naive term-by-term evaluation and is the basis of Taylor shifts P(x + h) in computer algebra.
- Why it matters in practice: Testing whether x = 2 is a root takes one short tableau instead of a full division. Chain it with candidates from the rational root theorem and each success deflates the polynomial, which is exactly how our factorization calculator works. For non-linear divisors, switch to long division.
The Formula
Formula / Method
- General form: For P(x) = aₙxⁿ + ... + a₁x + a₀ divided by (x − c), the tableau c | aₙ aₙ₋₁ ... a₀ produces quotient coefficients bₙ₋₁ ... b₀ and remainder r, where each bᵢ comes from multiply-by-c-and-add.
- Shortcut: The last entry is the remainder, which equals P(c) directly. Synthetic division therefore evaluates polynomials too, no separate substitution needed.
- Reading the result: The bottom row has one fewer coefficient group than the top row: those are the quotient's coefficients, and the final number is the remainder. A zero remainder means (x − c) is a factor.
- A common trap: The divisor must have the form (x − c). For (x + 3) use c = −3. For (2x − 1), factor out the 2 first and divide by (x − 1/2), then adjust the quotient.
The Method
How To Use
- Step 1. Write the dividend's coefficients in descending order, inserting 0 for any missing power of x.
- Step 2. Read c from the divisor (x − c), watching the sign carefully.
- Step 3. Press "Divide" and follow the tableau: bring down, multiply by c, add, repeat.
- Step 4. Read the quotient and remainder. If the remainder is zero, (x − c) is a factor, ready for the factorization calculator.
Key Features
- Displays the complete tableau with every multiply-and-add step shown.
- Automatically detects a zero remainder and flags the divisor as a factor.
- Inserts zero placeholders for missing terms so columns never shift.
- Reports P(c) as a byproduct, doubling as a fast evaluator.
- Chains divisions for cubics, deflating one degree at a time.
The Illustration
Visual Diagram
Example Concept
Divide x³ − 6x² + 11x − 6 by (x − 1), so c = 1. Coefficients: 1, −6, 11, −6.
Bring down 1. Multiply: 1 × 1 = 1. Add to −6: −5. Multiply: −5 × 1 = −5. Add to 11: 6. Multiply: 6 × 1 = 6. Add to −6: 0.
Bottom row: 1, −5, 6 | 0. Quotient x² − 5x + 6, remainder 0, so (x − 1) is a factor. Verify: (x − 1)(x² − 5x + 6) = x³ − 6x² + 11x − 6, the original polynomial.
The Deep Dive
Interactive Deep Dive
Synthetic division works because it is Horner's nested evaluation wearing a disguise. Writing P(x) = ((aₙx + aₙ₋₁)x + aₙ₋₂)x + ... + a₀ shows that evaluating at x = c needs only repeated multiply-by-c-and-add steps. The tableau performs exactly these steps while saving each intermediate result, and those saved intermediates turn out to be the quotient coefficients. Division and evaluation are the same computation viewed from two angles.
The last number of the tableau is the remainder, and the remainder theorem identifies it as P(c). This makes synthetic division two tools in one: a divider and an evaluator. Need P(7) for a degree-6 polynomial? Run the tableau with c = 7 and read the final entry. No powers to compute, no terms to add separately, just one clean cascade.
For a cubic, each successful division drops the degree by one, producing a deflation chain. Find one rational root with the rational root theorem, divide it out, and the remaining quadratic yields to the quadratic formula. Two tableaux can therefore solve a completely factorable cubic from scratch. This chain is precisely what the factorization calculator automates.
The tableau strictly requires a monic linear divisor (x − c). For (2x − 3), rewrite it as 2(x − 3/2): divide by (x − 3/2) with the tableau, then divide every quotient coefficient by 2. The remainder needs no adjustment. Forgetting this correction is the most common serious error, and for anything beyond linear divisors the honest tool is long division.
Paolo Ruffini published the method in 1804 and William Horner independently in 1819, but its real career began with computing. Horner's nested form needs only n multiplications for a degree-n polynomial instead of O(n²), and it rounds more gracefully than naive term-by-term evaluation. Every pocket calculator and math chip evaluates polynomials this way, so the tableau you draw by hand is the same algorithm silicon runs billions of times per second.
The Applications
Real-World Applications
🔎
Root Testing
Each rational candidate from the rational root theorem takes one short tableau to test. A zero in the last column confirms a root in seconds, which is why root hunting always pairs candidate generation with synthetic testing.
📝
Polynomial Reduction
After each confirmed root, the quotient row becomes the new dividend for the next round. This deflation cascade reduces a cubic to a quadratic mechanically, with no fresh long division at any stage.
🎓
Homework Efficiency
A full long division of a cubic fills half a page; the tableau fits in two lines. For homework and exams, that economy of writing also means fewer places for sign errors to hide.
🎨
Fast Function Evaluation
Building a table of values for graphing means evaluating P(c) repeatedly. Each evaluation is one tableau, far quicker than substituting into the expanded form term by term.
⚡
Numerical Methods
Iterative solvers like Newton's method need P(x) and P′(x) evaluated at every step. Horner-style evaluation, the engine inside synthetic division, is the standard fast and stable choice.
The Cautions
Common Mistakes to Avoid
MISTAKE Nº 1
Using the wrong sign for c: The divisor (x + 3) means c = -3, not 3. Read the divisor as (x - c) and solve for c every single time.
MISTAKE Nº 2
Forgetting zero placeholders: In x³ - 8, the x² and x terms are missing but their columns still exist. Write 1, 0, 0, -8, or every column shifts and the answer is garbage.
MISTAKE Nº 3
Applying to non-linear divisors: The tableau only works for divisors of the form (x - c). For quadratics or higher, use polynomial long division instead.
MISTAKE Nº 4
Misreading the quotient row: The bottom row has one fewer coefficient group than the top row. The last number is the remainder, not part of the quotient; its degree starts one lower than the dividend.
MISTAKE Nº 5
Claiming a factor from a nonzero remainder: Only a zero remainder proves (x - c) is a factor. A nonzero remainder is still useful, it equals P(c), but the division was not exact.
The Reference
Quick Reference Table
Quick Reference TableREF
Divisor Form(x − c) only
Remainder = 0c is a root, (x−c) is a factor
Remainder ≠ 0Remainder equals f(c)
Best forLinear divisors; much less writing than long division
OutputQuotient polynomial + remainder
The Collection
Explore Related Tools
The Questions
Frequently Asked Questions
What is synthetic division actually for? +
Dividing a polynomial by (x - c) quickly, and as a bonus, evaluating P(c). It replaces long division whenever the divisor is linear and monic, which covers the majority of hand computations.
When can I not use it? +
When the divisor is not of the form (x - c): quadratics, higher-degree divisors, or non-monic linears need adjustment. Use our long division calculator for those.
What does the last number mean? +
It is the remainder, and by the remainder theorem it equals P(c). Zero means (x - c) is a factor.
How do I divide by (x + 5)? +
Use c = -5. Always rewrite the divisor as (x - c) first and read c off carefully.
How do I divide by (3x - 2)? +
Factor out the 3 to get 3(x - 2/3), run the tableau with c = 2/3, then divide the quotient coefficients by 3.
Why must I insert zeros for missing terms? +
Every power of x owns a column. Skipping a missing term shifts all later columns left and corrupts the entire tableau.
Is it really faster than long division? +
Yes, for linear divisors it performs identical arithmetic with the x's stripped out, roughly half the writing and far fewer sign slips.
Can it find roots on its own? +
It tests candidates but does not invent them. Generate candidates with the rational root theorem, then let synthetic division judge each one.