Name: finding polynomial x-intercepts
Uploaded: 2025-11-26T05:43:55.000Z
Duration: 1 min 27 s
Channel: JoVE

Les zéros réels d’un polynôme sont les valeurs d’entrée pour lesquelles le polynôme est égal à zéro. Les zéros rationnels sont un sous-ensemble de zéros réels qui peuvent être exprimés sous la forme d’un rapport de deux entiers.Le théorème des zéros rationnels stipule que dans un polynôme à coefficients entiers, tout zéro rationnel doit être une fraction p sur q, où p divise le terme constant et q divise le coefficient principal.Le théorème des zéros rationnels identifie les zéros rationnels possibles en divisant les facteurs du terme constant, qui sont les nombres qui divisent uniformément la constante, par les facteurs du coefficient de la variable ayant la puissance la plus élevée.Ces valeurs possibles sont testées à l’aide d’une division synthétique, où les coefficients du polynôme sont divisés par chaque zéro réel candidat. Un reste non nul signifie que la valeur n’est pas un zéro du polynôme. Un reste de zéro confirme un zéro réel.Dans la conception d’emballages, le volume d’une boîte peut être modélisé par un polynôme en termes de taille de coupe. En fixant ce polynôme égal à un volume cible et en le réorganisant, on obtient une nouvelle équation égale à zéro. Les vrais zéros représentent les tailles de coupe qui permettent d’atteindre la capacité souhaitée sans gaspiller de matériau.

zéros réels des polynômes

Les polyn&#244;mes sont des expressions alg&#233;briques constitu&#233;es de termes dont les variables sont &#233;lev&#233;es &#224; des puissances enti&#232;res non n&#233;gatives. Un aspect central de l&#8217;analyse des fonctions polynomiales consiste &#224; d&#233;terminer leurs racines r&#233;elles (z&#233;ros r&#233;els), c&#8217;est-&#224;-dire les valeurs de la variable pour lesquelles le polyn&#244;me s&#8217;annule. Ces valeurs correspondent aux points d&#8217;intersection du graphe du polyn&#244;me avec l&#8217;axe des abscisses. 
Le th&#233;or&#232;me des racines rationnelles dresse la liste des racines rationnelles possibles d&#8217;une &#233;quation polynomiale &#224; coefficients entiers. Si f(x) = a_n x^n + &#8230; + a_0, alors toute racine rationnelle est de la forme p/q, o&#249; p divise le terme constant a_0 et q divise le coefficient de t&#234;te a_n. Ce th&#233;or&#232;me ne garantit pas que chaque candidat soit une racine, mais il fournit un ensemble fini de nombres rationnels &#224; tester.
Lorsqu&#8217;une valeur est v&#233;rifi&#233;e par la m&#233;thode de Ruffini et que le reste est nul, cette valeur est une racine du polyn&#244;me. Le polyn&#244;me peut alors &#234;tre factoris&#233; en cons&#233;quence, ce qui en r&#233;duit le degr&#233; et facilite l&#8217;analyse ult&#233;rieure. La r&#233;p&#233;tition de ce proc&#233;d&#233; simplifie le polyn&#244;me d&#8217;origine en facteurs de degr&#233; inf&#233;rieur.
Les z&#233;ros r&#233;els sont essentiels dans de nombreux domaines scientifiques et techniques. En physique, ils aident &#224; analyser le mouvement en identifiant les instants o&#249; un objet change de direction ou s&#8217;arr&#234;te. En images de synth&#232;se, ils servent &#224; la mod&#233;lisation de courbes et &#224; des transitions d&#8217;animation fluides. Savoir d&#233;terminer et interpr&#233;ter les z&#233;ros r&#233;els soutient une mod&#233;lisation, une analyse et une r&#233;solution de probl&#232;mes efficaces dans des applications vari&#233;es.

Zéros réels des polynômes

Les polyn&#244;mes sont des expressions alg&#233;briques constitu&#233;es de termes dont les variables sont &#233;lev&#233;es &#224; des puissances ...

Education

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JoVE Core

Math Fundamentals

Fondamentaux des mathématiques

Unit

Unité

Polynomial and Rational Functions

Fonctions polynomiales et rationnelles

KEY TERMS AND CONCEPTS

TERMES ET CONCEPTS CLÉS

Real Zeros of Polynomials

finding polynomial x-intercepts

Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis.

The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the leading coefficient an. The theorem does not prove that each candidate is a zero, but it does give a finite set of rational numbers to test.

Synthetic division is used to check whether a candidate value is a zero. If the remainder is zero, then the value is a real zero of the polynomial. The polynomial can then be factored, which lowers its degree and makes further analysis easier. Repeating this process breaks the polynomial into lower-degree factors.

Real zeros are useful in science and engineering. In physics, they help describe when an object stops or changes direction. In computer graphics, they help with curve modeling and smooth animation transitions. Knowing how to find and interpret real zeros supports modeling, analysis, and problem-solving in many fields.

Finding Polynomial X-Intercepts

quadratic models

Quadratic Motion and Parabolas

Quadratic models describe relationships where the rate of change changes at a constant rate. In math and science, this often shows up as a curved pattern called a parabola. These models help explain how a system changes over time or under a set force. A common example is the vertical motion of an object moving under gravity, such as a ball thrown into the air. The object rises to a maximum height and then falls back down. Gravity slows the ascent and speeds up the descent, which creates the...

introduction to polynomial functions

Polynomial Graphs, Zeros, and Degree

Polynomial functions are an important part of algebra and calculus. They are made from variables and constants combined by addition, subtraction, and multiplication. The variable is raised to a nonnegative integer exponent. A general polynomial function of degree n has a leading term aₙxⁿ, where aₙ is not equal to 0. The coefficient aₙ is the leading coefficient, and a₀ is the constant term. Polynomials are grouped by degree. A constant function has degree zero, a linear function has degree...

long division of polynomials

Polynomial Division: Quotient and Remainder

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synthetic disvision of polynomials

Using Synthetic Division to Find Factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

real zeros of polynomials

Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis. The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the...

fundamental theorem of algebra

Polynomial Roots in Complex Numbers

Polynomial roots in complex numbers are a key result in algebra. The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex zero. For a polynomial of degree n, where n is at least 1 and the leading coefficient a n is not 0, there is at least one solution in the complex number system. This result also applies to polynomials with real coefficients, because the real numbers are a subset of the complex numbers. The Complete...

complex zeros

Finding Polynomial Roots With Complex Numbers

Complex numbers help describe every root of a polynomial equation. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = -1. A complex zero is any value that makes P(x) = 0 for a polynomial P(x). The complex number system includes the real numbers, so it gives a complete way to study polynomial roots. Every polynomial of degree n  1 can be written as a product of n linear factors over the complex numbers. In that factorization, each c_i...

asymptotes in rational functions

Graphing Rational Functions with Asymptotes

Rational functions are quotients of two polynomials, with the denominator Q(x) not equal to zero. Their graphs often include asymptotes, which are lines the graph gets close to but does not touch. These lines help show how the function behaves near certain input values and at large values of x. A vertical asymptote appears when the denominator is zero and the numerator is not zero. At those x-values, the function is undefined. To find a vertical asymptote, solve Q(x) = 0. For example, a...

introduction to nonlinear inequalities

Graphing and Solving Nonlinear Inequalities

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application of nonlinear inequalities

Solving Nonlinear Inequalities on a Number Line

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side. After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and...

key terms and concepts

Learn how rational zeros, synthetic division, and factoring help identify polynomial x-intercepts and real zeros in high school algebra.

Finding Polynomial X-Intercepts | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Finding Polynomial X-Intercepts

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