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Kalkulator Bahagian Sintetik
Masukkan pekali polinomial anda di atas dan klik "Melaksanakan Bahagian Sintetik" untuk melihat keputusan.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 |
Tidak, pembahagian sintetik standard hanya berfungsi dengan sempurna untuk membahagi dengan binomial linear dalam bentuk<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x - c</span>.
ya. Jika kubik anda adalah<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x³ - 7x + 6</span>, anda mesti merawat<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>pekali sebagai 0. Alat ini secara automatik mengendalikan sifar yang dimasukkan.
Kemudian nombor yang anda uji bukan punca, tetapi selebihnya secara matematik mewakili penilaian<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">f(r)</span>.
Pembahagian sintetik ialah jalan pintas yang menggugurkan semua pembolehubah dan hanya berfungsi dengan pekali. Ia lebih cepat dan kurang terdedah kepada ralat untuk pembahagi linear, tetapi pembahagian panjang mengendalikan sebarang darjah pembahagi.
Ya! Jika bakinya ialah sifar selepas pembahagian sintetik, nombor yang anda uji sememangnya punca polinomial.
