Name: working with rational expression domains
Uploaded: 2025-11-26T05:44:35.000Z
Duration: 1 min 28 s
Channel: JoVE

In una città in crescita, l'espansione dei limiti e i nuovi sviluppi abitativi aumentano la popolazione, portando all'affollamento. La densità di popolazione quantifica quanto diventa affollata la città.Può essere modellato come un'espressione razionale in termini di tempo, t, una frazione dove il numeratore è la popolazione totale e il denominatore è l'area totale.Il dominio di questa espressione razionale si trova scomponendo il denominatore ed escludendo i valori che lo rendono zero.Questo può essere visualizzato su un grafico come interruzioni in cui la funzione diventa indefinita.La semplificazione di un'espressione razionale implica la scomposizione del numeratore e del denominatore e l'annullamento dei fattori comuni, solo quando il denominatore è diverso da zero.La moltiplicazione delle espressioni razionali implica la scomposizione di ogni espressione nella sua forma più semplice e quindi la moltiplicazione dei numeratori e dei denominatori insieme. L'espressione risultante viene semplificata annullando tutti i fattori comuni.L'aggiunta di queste espressioni richiede un denominatore comune, formato moltiplicando i denominatori e includendo i termini condivisi una sola volta. Ogni espressione viene riscritta con questa base e un numeratore regolato con i termini extra. Una volta allineati i denominatori, i numeratori vengono combinati per ottenere il risultato.

espressioni razionali

Le espressioni razionali sono frazioni algebriche in cui sia il numeratore che il denominatore sono polinomi. Queste espressioni seguono le regole aritmetiche delle frazioni numeriche, ma richiedono particolare attenzione a causa della presenza di variabili. Un aspetto fondamentale del lavoro con le espressioni razionali &#232; l'identificazione dei valori che rendono l'espressione indefinita, in genere quelli che danno luogo a una divisione per zero o a radici non definite.Determinazione del DominioIl dominio di un'espressione razionale include tutti i numeri reali tranne quelli che rendono il denominatore zero o violano i vincoli delle radici quadrate. Si consideri l'espressione:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40029/40029_Equation_1_1__editor_40029.svg' alt='40029_Equation_1_1_.svg' /></figure>Fattorizzazione del denominatore:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/120;' src='https://cloudfront.jove.com/files/ftp_upload/40029/40029_Equation_2_1__editor_40029.svg' alt='40029_Equation_2_1_.svg' /></figure>Il dominio esclude x=3 e x=-3, quindi il dominio &#232; l'insieme di tutti i numeri reali tranne questi valori.Semplificazione e MoltiplicazioneLa semplificazione implica la fattorizzazione sia del numeratore che del denominatore e l'eliminazione dei termini comuni. Ad esempio:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/250;' src='https://cloudfront.jove.com/files/ftp_upload/40029/40029_Equation_3_1__editor_40029.svg' alt='40029_Equation_3_1_.svg' /></figure>Per la moltiplicazione:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/40029/40029_Equation_4_1__editor_40029.svg' alt='40029_Equation_4_1_.svg' /></figure>Divisione e Addizione di Espressioni RazionaliPer dividere, si inverte la seconda espressione:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/40029/40029_Equation_5_4__editor_40029.svg' alt='40029_Equation_5_4_.svg' /></figure>Per l'addizione:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:1920/650;' src='https://cloudfront.jove.com/files/ftp_upload/40029/40029_Equation_6_1__editor_40029.svg' alt='40029_Equation_6_1_.svg' /></figure>Applicazioni e StrutturaLe espressioni razionali vengono utilizzate per modellare velocit&#224;, proporzioni e altre relazioni tra variabili. Le operazioni che coinvolgono queste espressioni compaiono in problemi relativi al moto, alla crescita, alla resistenza nei circuiti e all'ottimizzazione. Comprendere come semplificare, moltiplicare, dividere e combinare espressioni razionali &#232; fondamentale per la risoluzione di problemi in algebra e nelle scienze applicate.

Espressioni Razionali

Le espressioni razionali sono frazioni algebriche in cui sia il numeratore che il denominatore sono polinomi. Queste espressioni seguono le regole ...

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Math Fundamentals

Fondamenti di Matematica

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Foundations of Mathematics

Fondamenti di matematica

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

Rational Expressions

working with rational expression domains

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical.

To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator zero, and it also excludes values that break square root conditions. For example, after factoring a denominator, the excluded values may be x = 3 and x = -3, so the domain is all real numbers except those two values.

Simplifying a rational expression means factoring the numerator and denominator and canceling common factors. This is the same basic idea used with regular fractions. Multiplication uses the fraction rule as well. You multiply the numerators and the denominators, and then simplify when possible.

Division of rational expressions is handled by inverting the second expression and multiplying. Addition follows the usual fraction process, too. These skills matter in algebra and in real-world models that use rates, proportions, motion, growth, resistance in circuits, and optimization. Knowing how to simplify, multiply, divide, and combine rational expressions supports problem-solving in math and applied science.

Working with Rational Expression Domains

real number system

Number Types on the Real Line

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

real number operations

Real Numbers on the Number Line

Real numbers are the values that can be placed on a continuous number line. The system grew from counting numbers used for enumeration. It later expanded to include zero, which shows the absence of quantity, and negative numbers, which are opposites of counting numbers. The real number system also includes fractions and decimals that represent parts of a whole or equal divisions. When written in decimal form, these numbers may end or repeat in a regular pattern. Together, these values describe...

exponents

Exponent Rules and Scientific Notation

Exponents show repeated multiplication in a short form. They are important in algebra and in many other math topics, including scientific computation, scaling laws, and dimensional analysis. For a nonzero real number a and an integer n, a^n means a multiplied by itself n times. Exponent rules help simplify expressions with powers. These properties make it easier to work with exponential expressions in both symbolic and numerical form. They also support common calculations that use exponents in...

radicals

Simplifying Radicals

Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a. The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

algebraic expressions

Simplifying Algebraic Expressions

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

rational expressions

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical. To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

linear equations

Solving Linear Equations

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities. A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation...

quadratic equations

Solving Quadratics by Factoring and Formula

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems. The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions. In factored form, the...

complex numbers

Complex Numbers in Solving Equations

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1). A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

quadratic equations in the complex number system

quadratic-equations-in-the-complex-number-system

Complex Roots of Quadratic Equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

radical equations

Solving Radical Equations with Checks

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index. This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

inequalities

Solving Inequalities and Interval Notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

absolute value inequalities

Absolute Value Inequalities on a Number Line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable. An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

mathematical modeling: problem solving

Mathematical Modeling for Loan Payments

Mathematical modeling turns a real-life situation into an equation that can be solved. It starts with defining the problem, choosing measurable quantities, and assigning variables to them. Then a matching model is selected and the equation is solved for the unknown value. This process is especially useful in finance, where models help evaluate investments, loans, and repayment plans. One common example is finding the fixed monthly payment on a loan. That calculation uses the standard annuity...

key terms and concepts

Learn how to find the domain of rational expressions, simplify with factoring, and perform multiplication, division, and addition with algebraic fractions.

Working with Rational Expression Domains | 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

Working with Rational Expression Domains

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