Polynomial Solver: factor, divide, find roots

AskSia identifies the polynomial's form first. Always start by factoring out the greatest common factor (GCF) of all terms. After GCF, the method depends on the number of terms and the structure: two terms with subtraction often means difference of squares (a² minus b²) or difference of cubes (a³ minus b³); two terms with addition might be sum of cubes (a³ + b³); three terms (a trinomial) factors with the AC method or simple trinomial factoring; four terms usually factor by grouping. For higher-degree polynomials, the rational root theorem combined with synthetic division finds factors one at a time.

AskSia uses the rational root theorem to list candidate rational roots: any rational root p/q (in lowest terms) must have p as a factor of the constant term and q as a factor of the leading coefficient. Each candidate is tested by substitution. Once a root r is found, synthetic division divides the polynomial by (x minus r), reducing the degree by 1. The process repeats on the quotient until the polynomial is degree 2 (quadratic) or less. The remaining quadratic is solved with the quadratic formula, which may produce complex conjugate roots. All roots (real and complex) are reported with their multiplicities.

Polynomial long division works for any divisor (linear, quadratic, cubic, etc.) and follows the same layout as integer long division: divide leading terms, multiply back, subtract, bring down. Synthetic division is a faster shortcut that only works for divisors of the form (x minus c), where c is a constant. The synthetic division layout uses just the coefficients of the dividend and the value c. AskSia picks synthetic division automatically when the divisor is linear, and long division otherwise. Both methods produce the same quotient and remainder.

A root has multiplicity k if its factor appears k times in the factored form. For example, in p(x) = (x minus 2)³(x + 1), the root x = 2 has multiplicity 3 and the root x = -1 has multiplicity 1. Multiplicity affects how the polynomial graph behaves at the root: odd multiplicities cross the x-axis (with an inflection point for multiplicity 3 or higher), and even multiplicities touch the x-axis without crossing (the graph bounces off). AskSia identifies multiplicities explicitly when reporting roots.

AskSia is engineered for accuracy on standard high school and college coursework. Accuracy comes from subject-specialized models, a symbolic verification pass that catches arithmetic errors, and a self-check step that re-derives the answer before showing it to you.

Yes. After any solve, ask Sia to generate similar practice problems at SAT, ACT, AP, IB, or college difficulty, or build a flashcard set on the underlying concept in one tap. Useful for exam prep and spaced repetition before a quiz, midterm, or final.

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