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Syntetisk divisionsräknare
Ange dina polynomkoefficienter ovan och klicka på "Utför Synthetic Division" för att se resultat.Synthetic division is a streamlined algorithm for dividing a polynomial by a linear factor of the form (x − c). Instead of writing out full polynomial long division with variables and exponents, synthetic division uses only the coefficients arranged in a compact table, making it dramatically faster and less error-prone.
The process works by “cascading” multiplications and additions: bring down the leading coefficient, multiply by c, add to the next coefficient, multiply by c again, and repeat. The final number in the row is the remainder. If the remainder is zero, then c is a root and (x−c) is a factor. The remaining numbers form the quotient polynomial of one degree less.
Synthetic division serves dual purposes: division (finding the quotient when you know a factor) and evaluation (the remainder equals f(c) by the Remainder Theorem). This duality makes it an essential tool in the systematic root-finding process for cubics and higher-degree polynomials.
Quickly test whether a candidate value is a root by checking if the remainder is zero — much faster than substitution.
After finding one root, synthetic division reduces the cubic to a quadratic, enabling immediate use of the quadratic formula.
Students can verify factoring homework in seconds using the compact synthetic division format.
When dividing by (x + 3), the divisor value is −3, not +3. The sign convention often trips students up.
If a power is missing (e.g., no x² term), you MUST insert a 0 coefficient for that position.
Synthetic division only works for linear divisors (x − c). For quadratic or higher divisors, use polynomial long division.
| Divisor Form | (x − c) only |
| Remainder = 0 | c is a root, (x−c) is a factor |
| Remainder ≠ 0 | Remainder equals f(c) |
| Speed | ~3× faster than polynomial long division |
| Output | Quotient polynomial + remainder |
Nej, standard syntetisk division fungerar bara perfekt för att dividera med linjära binomialer i formen<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x - c</span>.
Ja. Om din kubik är<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x³ - 7x + 6</span>, måste du behandla<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>koefficient som 0. Verktyget hanterar automatiskt inmatade nollor.
Då är numret du testade inte en rot, utan resten representerar matematiskt utvärderingen<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">f(r)</span>.
Syntetisk division är en genväg som släpper alla variabler och bara fungerar med koefficienter. Den är snabbare och mindre felbenägen för linjära divisorer, men lång division hanterar vilken grad av divisor som helst.
Ja! Om resten är noll efter syntetisk division är talet du testade verkligen en rot av polynomet.
