Glossary

7.1 Multiply and Divide Rational Expressions

2 min read•june 24, 2024

Rational expressions are with algebraic terms. They're key in algebra, helping us solve complex problems and model real-world scenarios. Understanding how to work with them is crucial for tackling more advanced math concepts.

In this section, we'll learn to simplify, multiply, and divide rational expressions. We'll also explore rational functions and their operations. These skills are essential for solving equations and analyzing relationships between variables in various fields.

Rational Expressions

Undefined values in rational expressions

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  • undefined when equals zero
  • Find by setting denominator equal to zero and solving for
    • x+1x−2\frac{x+1}{x-2} undefined when x−2=0x-2=0
      • Solve x−2=0x-2=0
      • x=2x=2
      • Rational expression undefined when x=2x=2
  • The of a rational expression excludes values that make the denominator zero

Simplification of complex rationals

  • Factor and denominator completely
  • Cancel out common factors in numerator and denominator
  • Multiply remaining factors in numerator and denominator separately
  • Simplify x2−4x2+3x−10\frac{x^2-4}{x^2+3x-10}
    • Factor numerator: x2−4=(x+2)(x−2)x^2-4=(x+2)(x-2)
    • Factor denominator: x2+3x−10=(x+5)(x−2)x^2+3x-10=(x+5)(x-2)
    • Cancel out common factor (x−2)(x-2)
    • Simplified expression: x+2x+5\frac{x+2}{x+5}

Multiplication of rational expressions

  • Factor numerators and denominators of rational expressions
  • Multiply numerators together and denominators together
  • Cancel out common factors in resulting numerator and denominator
  • Multiply 2x+6x−1\frac{2x+6}{x-1} and x+43x+9\frac{x+4}{3x+9}
    • 2x+6x−1⋅x+43x+9=(2x+6)(x+4)(x−1)(3x+9)\frac{2x+6}{x-1} \cdot \frac{x+4}{3x+9} = \frac{(2x+6)(x+4)}{(x-1)(3x+9)}
    • Multiply numerators: (2x+6)(x+4)=2x2+14x+24(2x+6)(x+4)=2x^2+14x+24
    • Multiply denominators: (x−1)(3x+9)=3x2+6x−9(x-1)(3x+9)=3x^2+6x-9
    • Resulting product: 2x2+14x+243x2+6x−9\frac{2x^2+14x+24}{3x^2+6x-9}

Division of rational expressions

  • Rewrite division problem as multiplication problem by multiplying first rational expression by of second
    • Reciprocal of rational expression found by swapping numerator and denominator
  • Factor numerators and denominators, then multiply and cancel out common factors
  • Divide 3x2−122x+4\frac{3x^2-12}{2x+4} by x+1x−2\frac{x+1}{x-2}
    • 3x2−122x+4÷x+1x−2=3x2−122x+4⋅x−2x+1\frac{3x^2-12}{2x+4} \div \frac{x+1}{x-2} = \frac{3x^2-12}{2x+4} \cdot \frac{x-2}{x+1}
    • Factor numerators and denominators
      • 3x2−12=3(x2−4)=3(x+2)(x−2)3x^2-12=3(x^2-4)=3(x+2)(x-2)
      • 2x+4=2(x+2)2x+4=2(x+2)
    • Cancel out common factors and multiply remaining factors
      • 3x2−122x+4⋅x−2x+1=3(x+2)(x−2)2(x+2)⋅x−2x+1=3(x−2)2⋅x−2x+1=3(x−2)22(x+1)\frac{3x^2-12}{2x+4} \cdot \frac{x-2}{x+1} = \frac{3(x+2)(x-2)}{2(x+2)} \cdot \frac{x-2}{x+1} = \frac{3(x-2)}{2} \cdot \frac{x-2}{x+1} = \frac{3(x-2)^2}{2(x+1)}

Rational function operations

  • Rational functions are functions that can be written as ratio of two polynomials
  • To multiply or divide rational functions, follow same steps as multiplying or dividing rational expressions
  • Be cautious of undefined values when working with rational functions
    • Avoid inputting values that result in denominator being zero
  • Given rational functions f(x)=2x−1x+3f(x)=\frac{2x-1}{x+3} and g(x)=x2+4x+3x−1g(x)=\frac{x^2+4x+3}{x-1}, find f(x)⋅g(x)f(x) \cdot g(x)
    • f(x)⋅g(x)=2x−1x+3⋅x2+4x+3x−1f(x) \cdot g(x) = \frac{2x-1}{x+3} \cdot \frac{x^2+4x+3}{x-1}
    • Factor numerators and denominators
      • 2x−12x-1 and x+3x+3 cannot be factored further
      • x2+4x+3=(x+1)(x+3)x^2+4x+3=(x+1)(x+3)
    • Cancel out common factors and multiply remaining factors
      • 2x−1x+3⋅x2+4x+3x−1=2x−1x+3⋅(x+1)(x+3)x−1=(2x−1)(x+1)\frac{2x-1}{x+3} \cdot \frac{x^2+4x+3}{x-1} = \frac{2x-1}{x+3} \cdot \frac{(x+1)(x+3)}{x-1} = (2x-1)(x+1)
    • Resulting product: f(x)⋅g(x)=(2x−1)(x+1)f(x) \cdot g(x) = (2x-1)(x+1)

Components of Rational Expressions

  • Rational expressions are fractions containing
  • Numerator and denominator are polynomials, which are sums of terms with variables and
  • Variables in rational expressions represent unknown quantities
  • Exponents indicate how many times a base is multiplied by itself

Applying Rational Expressions

Key Terms to Review (22)

÷: The division symbol, also known as the obelus, is a mathematical operation that represents the division of one number by another. It is used to indicate that one quantity is to be divided by another, resulting in a quotient.
Algebraic Expressions: Algebraic expressions are mathematical representations that combine variables, numbers, and operations to represent quantitative relationships. They are the fundamental building blocks used in algebra to model and solve a wide range of problems.
Complex Fractions: A complex fraction is a fraction that has a fraction in the numerator, denominator, or both. They are used to represent and manipulate more complicated fractional expressions involving multiple levels of division.
Complex Rationals: Complex rationals are rational expressions where the numerator, denominator, or both contain complex numbers. These expressions involve the manipulation of complex fractions, requiring specific strategies for multiplication, division, and simplification.
Cross-Multiplication: Cross-multiplication is a fundamental algebraic technique used to solve equations and proportions. It involves multiplying the numerator of one fraction by the denominator of another fraction, and equating the resulting products to solve for an unknown variable.
Denominator: The denominator is the bottom number in a fraction, which represents the number of equal parts into which the whole has been divided. It plays a crucial role in various mathematical operations and concepts, including fractions, exponents, rational expressions, and rational inequalities.
Distributive Property: The distributive property is a fundamental algebraic rule that states that the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions involving multiplication and addition or subtraction.
Division of Rational Expressions: Division of rational expressions is the process of dividing one rational expression by another. This operation is essential in simplifying and manipulating rational expressions, which are fractions with polynomial numerators and denominators.
Domain: The domain of a function refers to the set of all possible input values for that function. It represents the range of values that the independent variable can take on, and it determines the set of values for which the function is defined.
Exponents: Exponents are a mathematical notation that indicate the number of times a base number is multiplied by itself. They represent repeated multiplication and are used to express very large or very small numbers concisely. Exponents are a fundamental concept in algebra and are crucial in understanding topics such as rational expressions, roots, and radical expressions.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. This technique is widely used in various areas of mathematics, including solving equations, simplifying rational expressions, and working with quadratic functions.
Fractions: A fraction is a numerical quantity that represents a part of a whole. It is expressed as a ratio of two numbers, the numerator and the denominator, which indicates how many equal parts the whole is divided into and how many of those parts are being considered.
Multiplication of Rational Expressions: Multiplication of rational expressions involves finding the product of two or more rational expressions, which are fractions with polynomial numerators and denominators. This operation is crucial in simplifying and manipulating algebraic expressions involving ratios of polynomials.
Numerator: The numerator is the part of a fraction that represents the number of equal parts being considered. It is the number above the fraction bar that indicates the quantity or number of units being referred to.
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 algebra and play a crucial role in various mathematical topics covered in this course.
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 fraction where the numerator and denominator are both polynomials, and it can be used to model and solve a variety of mathematical problems.
Rational Function: A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a function that can be written in the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomial functions and $Q(x)$ is not equal to zero.
Reciprocal: The reciprocal of a number is the value obtained by dividing 1 by that number. It represents the inverse or opposite of a quantity, and is a fundamental concept in various mathematical operations and applications.
Simplification: Simplification is the process of reducing or streamlining an expression, equation, or mathematical operation to its most basic or essential form, making it easier to understand, manipulate, or evaluate. This concept is central to various topics in mathematics, including fractions, rational expressions, radical expressions, and exponents.
Undefined Values: Undefined values refer to the concept in mathematics where a variable or expression cannot be assigned a specific numerical value. This often occurs when the denominator of a rational expression is zero, or when attempting to solve a rational equation that results in an invalid solution.
Variable: A variable is a symbol or letter that represents an unknown or changeable value in a mathematical expression, equation, or function. Variables are used to generalize and represent a range of possible values, allowing for the exploration of relationships and the solution of problems.
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 principle is fundamental in solving various algebraic equations and expressions involving polynomials, rational functions, and radicals.
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