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

In einer wachsenden Stadt erhöhen expandierende Grenzen und neue Wohnsiedlungen die Bevölkerung, was zu Überfüllungen führt. Die Bevölkerungsdichte quantifiziert, wie überfüllt die Stadt wird.Es kann als rationaler Ausdruck in Bezug auf die Zeit, t, modelliert werden, ein Bruch, wobei der Zähler die Gesamtbevölkerung und der Nenner die Gesamtfläche ist.Der Bereich dieses rationalen Ausdrucks wird gefunden, indem der Nenner faktorisiert und Werte ausgeschlossen werden, die ihn Null machen.Dies kann in einem Diagramm als Unterbrechungen visualisiert werden, bei denen die Funktion undefiniert wird.Bei der Vereinfachung eines rationalen Ausdrucks werden der Zähler und der Nenner faktorisiert und gemeinsame Teiler aufgehoben – nur wenn der Nenner nicht Null ist.Beim Multiplizieren rationaler Ausdrücke wird jeder Ausdruck in seine einfachste Form faktorisiert und dann die Zähler und Nenner miteinander multipliziert. Der resultierende Ausdruck wird vereinfacht, indem alle gemeinsamen Faktoren aufgehoben werden.Das Hinzufügen dieser Ausdrücke erfordert einen gemeinsamen Nenner, der durch Multiplizieren der Nenner gebildet wird und gemeinsame Begriffe nur einmal einschließen. Jeder Ausdruck wird mit dieser Basis neu geschrieben und mit einem Zähler versehen, der mit den zusätzlichen Begriffen angepasst wird. Sobald die Nenner ausgerichtet sind, werden die Zähler kombiniert, um das Ergebnis zu erhalten.

rationale ausdrücke

Rationale Ausdr&#252;cke sind algebraische Br&#252;che, bei denen sowohl Z&#228;hler als auch Nenner Polynome sind. Diese Ausdr&#252;cke folgen den arithmetischen Regeln von Zahlenbr&#252;chen, erfordern jedoch aufgrund der Variablen besondere Sorgfalt. Ein grundlegender Aspekt der Arbeit mit rationalen Ausdr&#252;cken ist die Ermittlung von Werten, die den Ausdruck undefiniert machen, typischerweise solche, die zu einer Division durch Null oder zu undefinierten Wurzelausdr&#252;cken f&#252;hren.Bestimmung des DefinitionsbereichsDer Definitionsbereich eines rationalen Ausdrucks umfasst alle reellen Zahlen mit Ausnahme derjenigen, bei denen der Nenner Null wird oder die gegen die durch Quadratwurzeln bedingten Einschr&#228;nkungen versto&#223;en. Betrachten Sie den Ausdruck:<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>Faktorisieren des Nenners:<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>Der Definitionsbereich schlie&#223;t x = 3 und x = &#8722;3 aus; er umfasst also alle reellen Zahlen mit Ausnahme dieser Werte.Vereinfachen und MultiplizierenBeim Vereinfachen werden Z&#228;hler und Nenner faktorisiert und gemeinsame Faktoren gek&#252;rzt. Beispiel:<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>F&#252;r die Multiplikation:<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>Division und Addition rationaler Ausdr&#252;ckeZum Dividieren invertieren Sie den zweiten Ausdruck:<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>F&#252;r die Addition:<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>Anwendungen und StrukturRationale Ausdr&#252;cke werden zur Modellierung von &#196;nderungsraten, Proportionen und anderen variablen Beziehungen verwendet. Rechenoperationen mit diesen Ausdr&#252;cken treten bei Aufgaben zur Bewegung, zum Wachstum, zu Widerst&#228;nden in Schaltkreisen und zur Optimierung auf. Das Verst&#228;ndnis der Verfahren zum Vereinfachen, Multiplizieren, Dividieren und Kombinieren rationaler Ausdr&#252;cke ist grundlegend f&#252;r die Probleml&#246;sung in der Algebra und in den angewandten Wissenschaften.

Rationale Ausdrücke

Rationale Ausdr&#252;cke sind algebraische Br&#252;che, bei denen sowohl Z&#228;hler als auch Nenner Polynome sind. Diese Ausdr&#252;cke folgen den ...

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KEY TERMS AND CONCEPTS

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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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