Enter your dividend polynomial (e.g. x^4 - 5x^3 + 7x^2 - 34x - 1) and a linear divisor (e.g. x - 5) to perform synthetic division. The Synthetic Division Calculator returns the quotient polynomial, the remainder, and a full step-by-step breakdown of the synthetic division table.
Quotient Polynomial
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Remainder
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Result Expression
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Synthetic division is a shorthand method of dividing a polynomial by a linear binomial of the form (x − c). Instead of writing out full polynomial long division, you work only with the coefficients, making the process faster and less error-prone.
First, write the root 'c' from your divisor (x − c) on the left. Then list all coefficients of the dividend in order, including zeros for missing terms. Bring down the first coefficient, multiply it by c, add the result to the next coefficient, and repeat until you reach the last coefficient. The final number is the remainder, and the preceding numbers are the coefficients of the quotient.
Use synthetic division when the divisor is a linear binomial (degree 1) such as x − 3 or 2x + 1 — it is faster and neater. Use polynomial long division when the divisor has degree 2 or higher, since synthetic division only works for linear divisors.
Yes. The relationship is expressed as P(x) = (x − c) · Q(x) + R, where P(x) is the dividend polynomial, (x − c) is the linear divisor, Q(x) is the quotient polynomial, and R is the remainder. If R = 0, then (x − c) is a factor of P(x).
By the Factor Theorem, if you divide P(x) by (x − c) and the remainder is 0, then c is a root (zero) of P(x). Synthetic division is therefore a quick way to test potential roots and to factor higher-degree polynomials.
When the divisor is of the form ax + b (where a ≠ 1), you first find the root c = −b/a. You can still use synthetic division with c, but you must then divide each coefficient of the resulting quotient by 'a' at the end to account for the leading coefficient.
If your dividend is missing a term (for example, x^3 + 2x − 5 has no x^2 term), you must insert a 0 as a placeholder coefficient for that missing term. This ensures the synthetic division table aligns correctly with every power of x.
Yes. The synthetic division method works for any real coefficients, including decimals and fractions. Just enter them as decimal numbers (e.g. 0.5 instead of 1/2) and the calculator will process them correctly.