Glossary

1.6 Rational Expressions

3 min read•june 24, 2024

Rational expressions are like mathematical fractions on steroids. They involve polynomials in both the top and bottom parts. Simplifying these expressions is crucial for solving complex math problems and understanding their behavior.

Operations with rational expressions follow similar rules to regular fractions, but with a twist. , , , and all require special techniques. Solving equations with rationals and analyzing complex rational expressions are advanced skills that build on these fundamentals.

Simplifying and Operating on Rational Expressions

Simplification of rational expressions

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  • Factor numerator and denominator completely
    • Find (GCF) of terms in numerator and denominator
    • Factor out GCF from both numerator and denominator
    • Factor remaining expressions using techniques like:
      • : a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
      • Perfect square trinomials: a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a+b)^2 and a2−2ab+b2=(a−b)2a^2 - 2ab + b^2 = (a-b)^2
      • : ac+ad+bc+bd=(a+b)(c+d)ac + ad + bc + bd = (a+b)(c+d)
  • Cancel common factors in numerator and denominator
    • Find factors appearing in both numerator and denominator
    • Divide out common factors to simplify

Operations with rational expressions

  • Multiplication of rational expressions
    • Multiply numerators together for new numerator
    • Multiply denominators together for new denominator
    • Simplify resulting by and canceling common factors
  • Division of rational expressions
    • Rewrite division as multiplication by of divisor
      • Reciprocal of ab\frac{a}{b} is ba\frac{b}{a}
    • Multiply numerators together for new numerator
    • Multiply denominators together for new denominator
    • Simplify resulting rational expression by and canceling common factors

Addition and subtraction of rationals

  • Adding and subtracting rational expressions with
    • Keep common denominator
    • Add or subtract numerators
    • Simplify resulting rational expression by factoring and canceling common factors
  • Adding and subtracting rational expressions with
    • Find (LCD) of rational expressions
      • LCD is (LCM) of denominators
    • Rewrite each rational expression with LCD as denominator
      • Multiply numerator and denominator of each expression by factor needed to obtain LCD
    • Add or subtract numerators of equivalent expressions
    • Simplify resulting rational expression by factoring and canceling common factors

Solving Equations and Analyzing Complex Rational Expressions

Equations with rational expressions

  • Clear denominators by multiplying both sides of equation by LCD of all rational expressions
  • Simplify resulting equation by distributing and combining like terms
  • Solve simplified equation using appropriate techniques like:
    • Factoring and applying
    • Using : x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a} for equations in form ax2+bx+c=0ax^2 + bx + c = 0
  • Check solutions by substituting back into original equation
    • Reject solutions that result in denominator equal to zero (extraneous)

Analysis of complex rationals

  • Identify main bar and consider expressions above and below it as separate units
  • Simplify numerator and denominator separately by factoring and canceling common factors
  • If numerator or denominator contains fractions, use techniques for adding, subtracting, multiplying, or dividing rational expressions to simplify
  • Combine simplified numerator and denominator to form simplified
  • If necessary, repeat process until expression cannot be simplified further

Key Concepts in Rational Expressions

  • : The set of all possible input values for which a rational expression is defined
  • : A line that the graph of a rational function approaches but never crosses
  • : An expression consisting of variables and coefficients involving only addition, subtraction, and multiplication operations
  • Rational expressions are fractions where both numerator and denominator are polynomials

Key Terms to Review (38)

Addition: Addition is a fundamental mathematical operation that combines two or more quantities to find their sum. It is one of the basic arithmetic operations, along with subtraction, multiplication, and division, and is essential in the context of rational expressions.
Associative property of multiplication: The associative property of multiplication states that the way in which numbers are grouped in a multiplication problem does not change the product. Mathematically, for any real numbers $a$, $b$, and $c$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
Asymptote: An asymptote is a line or curve that a graph approaches but never touches. It represents the limit of a function's behavior as the input variable approaches a particular value. Asymptotes are an important concept in various mathematical topics, including rational expressions, functions, rational functions, exponential functions, logarithmic functions, and exponential and logarithmic models.
Cancellation: Cancellation is the process of eliminating common factors or variables from the numerator and denominator of a rational expression, which simplifies the expression and makes it easier to evaluate. This concept is particularly important in the context of 1.6 Rational Expressions, as it allows for the simplification of complex fractions and the efficient manipulation of algebraic expressions involving rational functions.
Complex Rational Expression: A complex rational expression is a rational expression that contains one or more variables in both the numerator and denominator. It is a fraction where both the numerator and denominator are polynomial expressions with variables.
Difference of squares: The difference of squares is a specific type of polynomial that takes the form $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. It is based on the property that the product of a sum and a difference of two terms results in the difference of their squares.
Difference of Squares: The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.
Division: Division is the mathematical operation of splitting a quantity into equal parts or groups. It is the inverse of multiplication and is used to determine how many times one number is contained within another.
Division Algorithm: The Division Algorithm for polynomials states that given any two polynomials, a dividend and a non-zero divisor, there exist unique quotient and remainder polynomials. The degree of the remainder polynomial is less than the degree of the divisor.
Domain: The domain of a function is the complete set of possible input values (x-values) that allow the function to work within its constraints. It specifies the range of x-values for which the function is defined.
Domain: The domain of a function refers to the set of input values for which the function is defined. It represents the range of values that the independent variable can take on, and it is the set of all possible values that can be plugged into the function to produce a meaningful output.
Extraneous solution: An extraneous solution is a solution derived from an equation that is not valid within the original equation. Extraneous solutions often arise when both sides of an equation are manipulated.
Extraneous Solution: An extraneous solution is a solution to an equation that does not satisfy the original constraints or conditions of the problem. It is a solution that is mathematically valid but does not make sense in the context of the problem being solved.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Fraction: A fraction is a mathematical representation of a part of a whole. It is used to express the relationship between a numerator, which represents the number of parts, and a denominator, which represents the total number of equal parts that make up the whole.
Greatest common factor: The greatest common factor (GCF) of two or more numbers is the largest number that divides each of them without leaving a remainder. It is useful in simplifying fractions, factoring polynomials, and solving equations.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides two or more integers without a remainder. It is a fundamental concept in algebra that is essential for understanding real numbers, factoring polynomials, working with rational expressions, solving quadratic equations, and analyzing power and polynomial functions.
Grouping: Grouping is the process of combining or organizing mathematical expressions, functions, or elements into a single unit to simplify operations, enhance readability, or perform specific calculations. It is a fundamental concept in mathematics that is particularly relevant in the contexts of rational expressions and the graphs of polynomial functions.
Least common denominator: The least common denominator (LCD) of two or more rational expressions is the smallest positive integer that is divisible by each of their denominators. It is essential for adding, subtracting, or comparing fractions.
Least Common Denominator: The least common denominator (LCD) is the smallest positive integer that is divisible by all the denominators in a set of fractions. It is a crucial concept in rational expressions, as it allows for the comparison and manipulation of fractions with different denominators.
Least Common Multiple: The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers. It is a fundamental concept in mathematics, particularly in the context of rational expressions, where it is used to find a common denominator for fractions with different denominators.
Like Denominators: Like denominators refer to fractions or rational expressions that have the same denominator. This concept is crucial in the context of 1.6 Rational Expressions, as it allows for the simplification and manipulation of these expressions through common denominators.
Multiplication: Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is a fundamental concept that is essential in various areas of mathematics, including algebra, rational expressions, rational functions, and the polar form of complex numbers.
Perfect square trinomial: A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
Perfect Square Trinomial: A perfect square trinomial is a special type of polynomial expression that can be factored as the square of a binomial. It is a three-term polynomial in the form $a^2 + 2ab + b^2$, where $a$ and $b$ are real numbers.
Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents. It can be written in the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n, a_{n-1}, ..., a_1, a_0$ are constants and $n$ is a non-negative integer.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers. Polynomials are fundamental in various areas of mathematics, including algebra, calculus, and the study of functions.
Quadratic formula: The quadratic formula is used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Rational expression: A rational expression is a fraction where both the numerator and the denominator are polynomials. The denominator cannot be zero.
Rational Expression: A rational expression is a mathematical expression that consists of one or more polynomials divided by one or more polynomials. It represents a ratio of two polynomial functions and can be used to model and analyze various mathematical relationships.
Reciprocal: The reciprocal of a number is the value obtained by dividing 1 by that number. It represents the inverse relationship between two quantities, where one quantity is the multiplicative inverse of the other.
Reciprocal function: A reciprocal function is a function of the form $f(x) = \frac{1}{g(x)}$, where $g(x)$ is a non-zero polynomial. The simplest example is $f(x) = \frac{1}{x}$.
Simplification: Simplification is the process of reducing the complexity of an expression or equation by applying various mathematical rules and techniques to obtain a simpler, more manageable form. This concept is crucial in the context of rational expressions, rational functions, trigonometric identities, and trigonometric expressions, as it allows for more efficient calculations and better understanding of the underlying mathematical relationships.
Subtraction: Subtraction is a fundamental mathematical operation that involves the removal of one number or quantity from another. It is used to find the difference between two values and is an essential skill in various mathematical contexts, including the study of rational expressions.
Unlike Denominators: Unlike denominators refer to the situation where two or more fractions have different denominators. This concept is crucial when performing operations such as addition or subtraction on rational expressions, as it requires a common denominator to combine the fractions effectively.
Zero-Product Property: The zero-product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is particularly important in the context of rational expressions, where it is used to solve equations and simplify expressions.
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