--- title: Finding Zeros of a Polynomial - Methods & Techniques description: Finding zeros of a polynomial means locating the roots. Factoring and the rational root theorem, explained step by step. url: https://astra-ai.co/videos/polynomials/finding-zeros-of-a-polynomial/ locale: en type: video-chapter keywords: - finding zeros of a polynomial - rational root theorem - factoring - polynomial roots published: 2026-10-01T09:50:33.572Z updated: 2026-10-04T21:58:54.635Z alternates: bg: https://astra-ai.co/bg/videos/polinomi/namirane-na-nuli-na-polinom/ br: https://astra-ai.co/br/videos/polinomios/capitulo-matematica-026/ cs: https://astra-ai.co/cs/videos/polynomy/hledani-nul-polynomu/ da: https://astra-ai.co/da/videos/polynomier/bestemmelse-af-polynomiets-nulpunkter/ de: https://astra-ai.co/de/videos/polynome/nullstellen-von-polynomen-finden/ el: https://astra-ai.co/el/videos/polyonima/eyresi-rizon-polyonymou/ en: https://astra-ai.co/videos/polynomials/finding-zeros-of-a-polynomial/ en-IN: https://astra-ai.co/en-IN/videos/polynomials/chapter-af1c1b9d/ en-PH: https://astra-ai.co/en-PH/videos/polynomials/chapter-af1c1b9d/ es: https://astra-ai.co/es/videos/polinomios/ceros-de-un-polinomio-1/ es-AR: https://astra-ai.co/es-AR/videos/polinomios/capitulo-20/ es-CL: https://astra-ai.co/es-CL/videos/polinomios/capitulo-20/ es-PE: https://astra-ai.co/es-PE/videos/polinomios/capitulo-video-af1c1b9d/ et: https://astra-ai.co/et/videos/polunoomid/polunoomi-nullkohtade-leidmine/ fi: https://astra-ai.co/fi/videos/polynomit/polynomin-nollakohtien-etsiminen/ fr: https://astra-ai.co/fr/videos/polynomes/trouver-les-zeros-dun-polynome/ hr: https://astra-ai.co/hr/videos/polinomi/trazenje-nula-polinoma/ hu: https://astra-ai.co/hu/videos/polinomok/polinom-zerushelyeinek-meghatarozasa/ id: https://astra-ai.co/id/videos/polinom/menentukan-nol-polinom/ it: https://astra-ai.co/it/videos/polinomi/trovare-zeri-polinomio/ ja: https://astra-ai.co/ja/videos/polynomials-ja/ja-chapter-20/ km: https://astra-ai.co/km/videos/polynomials/chapter-af1c1b9d/ ko: https://astra-ai.co/ko/videos/polynomials-ko/20/ lo: https://astra-ai.co/lo/videos/lo-polynomials/lo-af1c1b9d/ lt: https://astra-ai.co/lt/videos/polinomai/polinomo-saknu-radimas/ lv: https://astra-ai.co/lv/videos/polinomi/polinoma-saknu-atrasana/ ms: https://astra-ai.co/ms/videos/polynomials/bab-26-matematik-asas/ ms-BN: https://astra-ai.co/ms-BN/videos/polinomial/bab-matematik-20/ mx: https://astra-ai.co/mx/videos/polinomios/capitulo-20/ my: https://astra-ai.co/my/videos/my-polynomials/my-af1c1b9d/ nl: https://astra-ai.co/nl/videos/veeltermen/nulpunten-van-een-polynoom-vinden/ no: https://astra-ai.co/no/videos/polynomer/finne-nullpunkter-til-et-polynom/ pl: https://astra-ai.co/pl/videos/wielomiany/wyznaczanie-miejsc-zerowych-wielomianu/ pt: https://astra-ai.co/pt/videos/polinomios/encontrando-os-zeros-de-um-polinomio/ pt-TL: https://astra-ai.co/pt-TL/videos/polinomios/capitulo-de-video-020/ ro: https://astra-ai.co/ro/videos/polinoame/determinarea-radacinilor-unui-polinom/ sk: https://astra-ai.co/sk/videos/polynomy/kapitola-020/ sl: https://astra-ai.co/sl/videos/polinomi/iskanje-nicel-polinoma/ sr: https://astra-ai.co/sr/videos/polinomi/poglavlje-020/ sv: https://astra-ai.co/sv/videos/polynom/bestamning-av-polynoms-nollstallen/ th: https://astra-ai.co/th/videos/phahunam/bot-26-harang-phahunam/ tr: https://astra-ai.co/tr/videos/polinomlar/bolum-020/ uk: https://astra-ai.co/uk/videos/polynomials-uk/uk-chapter-26/ vi: https://astra-ai.co/vi/videos/vi-topic-01/vi-chapter-26/ zh-HK: https://astra-ai.co/zh-HK/videos/shuxue-kecheng-001/shuxue-zhang-026/ zh-TW: https://astra-ai.co/zh-TW/videos/polynomials/chapter-af1c1b9d/ --- # Finding Zeros of a Polynomial > Finding zeros of a polynomial means locating the inputs that make it zero. This chapter uses factoring and the rational root theorem. ## INTRODUCTION AND GOAL Finding the zeros of a polynomial means solving the equation P(x) = 0, where P(x) is a polynomial with real or rational coefficients. The goal is to determine all values of x for which the polynomial has a value of 0. These values are key for factorization, graphical representation, and further algebraic procedures such as division and simplification of expressions. A polynomial of degree 'n' has at most 'n' real or complex zeros, counting multiplicities. ## METHODS FOR FINDING ZEROS - **FACTORING OUT THE GREATEST COMMON FACTOR (GCF)** If all terms have a common factor, first factor it out to simplify the polynomial. Example: P(x) = x³ – x² = x²(x – 1) → Zeros: x = 0 (with multiplicity 2), x = 1. - **FACTORING THE POLYNOMIAL** If the polynomial can be rewritten as a product of lower-degree factors, the zeros can be found from each factor. Example: P(x) = (x – 2)(x + 3)(x – 1) → Zeros: x = 2, x = –3, x = 1. - **RATIONAL ROOT THEOREM (FOR POLYNOMIALS WITH INTEGER COEFFICIENTS)** If a polynomial with integer coefficients has rational zeros, they are of the form: x = ±p/q, where 'p' is a divisor of the constant term (a₀), and 'q' is a divisor of the leading coefficient (aₙ). Example: P(x) = x³ – 2x² – 5x + 6 Possible rational zeros (divisors of 6 divided by divisors of 1): ±1, ±2, ±3, ±6. By testing: P(1) = 1³ – 2(1)² – 5(1) + 6 = 1 – 2 – 5 + 6 = 0 → x = 1 is a zero. - **HORNER'S ALGORITHM (SYNTHETIC DIVISION)** Once one zero 'c' is known, the polynomial can be divided by (x – c) using Horner's algorithm (or synthetic division) to find the remaining quotient, whose zeros can then be sought. Example: P(x) = x³ – 6x² + 11x – 6 Test P(1) = 1³ – 6(1)² + 11(1) – 6 = 1 – 6 + 11 – 6 = 0 → x = 1 is a zero. Divide P(x) by (x – 1): The quotient is x² – 5x + 6. Factoring the quadratic: x² – 5x + 6 = (x-2)(x-3) → Zeros of the quotient are x = 2, x = 3. Final zeros: x = 1, x = 2, x = 3. - **FORMULAS FOR SPECIAL CASES** - - **QUADRATIC POLYNOMIAL:** P(x) = ax² + bx + c → Zeros using the quadratic formula: x = (–b ± √(b² – 4ac)) / 2a - **CUBIC OR HIGHER-DEGREE POLYNOMIALS:** If factoring or the rational root theorem is not straightforward, numerical methods or more advanced techniques might be required. - **GRAPHICAL METHOD** Zeros can also be approximated by looking at the graph of the function f(x) = P(x) – the points where the graph intersects the x-axis correspond to the real zeros. ## CONCLUSION Finding the zeros of a polynomial is a systematic process based on factoring, testing rational values, division, and using known formulas. Knowledge of methods for finding zeros allows for understanding the structure of a polynomial, its factorization, and preparation for solving more complex equations.