Synthetic Division Calculator

Enter a polynomial such as x^3 − 6x^2 + 11x − 6, or paste its coefficients. Divide by x − r or a general linear divisor ax + b and get the exact quotient, remainder, classic synthetic tableau, Remainder Theorem check, and full reconstruction.

Synthetic division workbench

Type the polynomial the way you see it. Get the exact tableau, quotient and proof.

Use a polynomial expression or coefficient list, then divide by x − r or a general linear divisor ax + b. Exact rational arithmetic keeps fractions exact instead of hiding rounding inside the steps.

Exact fractionsClassic tableauP(r) checkExact reconstructionRuns locally

Dividend polynomial or coefficientsExamples: x^4 − 6x^2 − 30 or 1, 0, −6, 0, −30. Missing powers are filled automatically for polynomial input.Linear divisorUse x − 2, x + 3, 2x − 1, or just 2 to mean x − 2.

Quotient—

—

—Remainder

—Synthetic value r

—Is divisor a factor?

—Remainder Theorem

Synthetic division tableauEnter a valid problem to build the table.

Waiting for input.

Ready.

Step-by-step multiply and add

Recent calculations & privacy

Saved calculations are optional and stay in local browser storage. Portable links put only the current polynomial and divisor into the URL hash; no calculation is uploaded to WebToolArc.

Scope: this calculator is intentionally for one-variable polynomials divided by a linear divisor. It accepts exact integers, decimals and simple fractions. For higher-degree divisors, use polynomial long division. A zero remainder proves the linear divisor is a factor of the entered polynomial; it does not prove that every coefficient or divisor was entered as intended.

What this workbench verifies beyond the answer

Missing powers are handled

Expression input such as x^4 − 6x^2 − 30 automatically inserts zero coefficients for x^3 and x. Coefficient-list input still requires those zeros explicitly, which mirrors classroom notation.

General ax + b divisors work

A divisor such as 2x − 1 is normalized to x − 1/2 for the synthetic row, then the quotient is scaled by the divisor's leading coefficient. The remainder remains P(1/2).

The whole identity is checked

After division, WebToolArc reconstructs dividend = divisor × quotient + remainder with exact rational coefficients. This catches more than checking the final remainder alone.

Algebra round-trip cluster

Solve once, then verify the same object through another representation

Independent round trips and invariants expose sign, scaling, reconstruction and degeneracy mistakes that a single formatted answer can hide.

Quadratics Matrices Systems

All 28 algebra tools

Matrix InverseA↔A⁻¹ identity round trips MidpointEndpoint reflection round trip Linear EquationScale invariance + degeneracy classes DeterminantTranspose/row-operation invariants Quadratic VertexStandard ↔ vertex coefficient round trip Slope InterceptPoints ↔ line-form equivalence Polynomial DivisionQuotient/remainder reconstruction Standard FormSlope-intercept ↔ standard round trip

Transform, then verify

Equivalent forms should agree. Use discriminants, roots, factoring, vertex form, substitution, matrix checks, or the Remainder Theorem where appropriate to verify the result rather than relying on a single opaque output.

Remainder Theorem made visible

The final synthetic value is cross-checked against P(r), making the x − r divisor relationship explicit.

Scope and precision

These are bounded browser calculators, not a universal computer algebra system. WebToolArc keeps exact algebraic identities visible where practical, uses stable floating-point algorithms for numeric work, distinguishes undefined/no-solution/infinite-solution cases, and shows an independent check after the main Calculate action.

Practical guide and verification

Use the tool first, then apply these checks to verify the inputs, interpret the result, and hand it off without displacing the primary workflow.

Confirm the divisor is linear in the required form

Synthetic division is a shortcut for division by a linear factor such as x minus r. If the divisor has another leading coefficient or higher degree, transform it correctly or use polynomial long division. Entering the wrong r can produce a tidy table for the wrong divisor.

Include zero coefficients for missing powers

A polynomial that skips a power still needs a placeholder coefficient in the synthetic row. For x cubed plus 2x minus 5, the x squared coefficient is zero. Omitting that zero shifts every later coefficient and changes both the quotient and remainder.

Use the remainder theorem as an independent check

For division by x minus r, the synthetic remainder should equal the original polynomial evaluated at r. Compare those two numbers before accepting the quotient. A mismatch is a strong signal that the coefficient order, missing-power placeholders, or divisor sign is wrong.

Reconstruct the dividend after division

Multiply the displayed quotient by the divisor and add the remainder. The coefficients should reproduce the original polynomial. This reverse check validates the whole transformation rather than only the final remainder.