Penyelesai persamaan padu khusus dengan punca sebenar dan kompleks, langkah kaedah Cardano, grafik padu dan contoh yang berfungsi.
Kalkulator Pemfaktoran Polinomial
Masukkan pekali polinomial anda di atas dan klik "Faktorkan Polinomial" untuk melihat keputusan.Polynomial factorization is the process of breaking a cubic expression into a product of simpler factors. For a cubic ax³ + bx² + cx + d, the ideal factored form is a(x − r₁)(x − r₂)(x − r₃), where r₁, r₂, r₃ are the roots. Factoring transforms solving the equation into a straightforward zero-product problem.
Common factoring strategies for cubics include: common factor extraction (pulling out shared terms), grouping (splitting into pairs that share a binomial factor), sum/difference of cubes (x³ ± a³), and rational root testing followed by synthetic division. When a rational root r is found, dividing by (x − r) reduces the cubic to a quadratic, which the quadratic formula handles.
Factorization is more than just solving equations — it reveals the structure of a polynomial. Factors expose symmetries, shared roots with other polynomials, and simplification opportunities in rational expressions. In computer algebra systems, efficient factorization algorithms are fundamental to symbolic mathematics.
Factoring is the fastest path to roots when rational factors exist. It avoids the complexity of Cardano's method entirely.
Factored forms enable cancellation in rational expressions, essential for calculus limits and integration.
Symbolic math engines use factorization as a core operation for polynomial GCD, simplification, and integration.
Many cubics have irrational or complex roots and cannot be factored using integers alone. Use Cardano's method as a fallback.
The factored form is a(x−r₁)(x−r₂)(x−r₃), not just (x−r₁)(x−r₂)(x−r₃). Don't forget the 'a' out front.
The Rational Root Theorem generates a list of candidates. You must test ALL of them before concluding no rational root exists.
| Goal | a(x − r₁)(x − r₂)(x − r₃) |
| Sum of Cubes | a³+b³ = (a+b)(a²−ab+b²) |
| Diff of Cubes | a³−b³ = (a−b)(a²+ab+b²) |
| Strategy | Find 1 root → divide → quadratic formula |
| Verification | Expand factors to confirm original polynomial |
Tidak, banyak kubik dunia sebenar tidak boleh difaktorkan dengan bersih ke dalam integer atau pecahan piawai, yang memerlukan kaedah berangka.
Alat meninggalkannya dalam format<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">(x - r)(ax² + bx + c)</span>mewakili bahagian akar kompleks.
Ya, jika nisbah sepadan, pengumpulan ialah cara terpantas mutlak untuk menyelesaikan kubik dengan tangan.
Ia ialah kaedah di mana anda membahagikan kubik empat penggal kepada kumpulan dua penggal dan mencari faktor binomial sepunya. Jika kedua-dua kumpulan berkongsi faktor yang sama, faktor padu dengan kemas.
Cuba pemfaktoran dahulu — ia lebih mudah dan lebih pantas apabila ia berfungsi. Jika tiada akar rasional wujud atau pengelompokan gagal, maka kaedah Cardano adalah sandaran yang boleh dipercayai.
