Glossary

7.2 Add and Subtract Rational Expressions

3 min read•june 24, 2024

Rational expressions are fractions with algebraic terms. Adding and subtracting them requires finding common denominators and combining numerators. This process is similar to working with regular fractions, but with variables involved.

Understanding these operations is crucial for solving complex equations and . Mastering this skill helps in various math and science applications, from basic algebra to advanced calculus and physics.

Adding and Subtracting Rational Expressions

Addition of rational expressions

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  • Add rational expressions with common denominators by combining the numerators using addition while keeping the the same
    • 37+27=3+27=57\frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7}
    • 4xy2+7xy2=4x+7xy2=11xy2\frac{4x}{y^2} + \frac{7x}{y^2} = \frac{4x+7x}{y^2} = \frac{11x}{y^2}
  • Simplify the resulting by combining like terms or performing any necessary arithmetic operations, but leave the denominator unchanged
  • This process applies to various types of rational expressions, including algebraic expressions and

Least common denominator identification

  • To add or subtract rational expressions with unlike denominators, determine the (LCD) first
    • LCD is the (LCM) of the individual denominators
    • For 23\frac{2}{3} and 56\frac{5}{6}, the LCD is 6 (LCM of 3 and 6)
    • For x2y\frac{x}{2y} and 34y\frac{3}{4y}, the LCD is 4y (LCM of 2y and 4y)
  • Multiply both the numerator and denominator of each rational expression by the factor needed to obtain the LCD
    • To add 14\frac{1}{4} and 36\frac{3}{6}, multiply the first expression by 33\frac{3}{3} to get 1⋅34⋅3=312\frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12} and the second expression by 22\frac{2}{2} to get 3⋅26⋅2=612\frac{3 \cdot 2}{6 \cdot 2} = \frac{6}{12}

Subtraction with unlike denominators

  • After finding the LCD and rewriting the rational expressions with this , subtract the numerators and keep the LCD as the denominator
    • 56−13=1012−412=10−412=612=12\frac{5}{6} - \frac{1}{3} = \frac{10}{12} - \frac{4}{12} = \frac{10-4}{12} = \frac{6}{12} = \frac{1}{2}
    • 3x4y−2x3y=9x12y−8x12y=9x−8x12y=x12y\frac{3x}{4y} - \frac{2x}{3y} = \frac{9x}{12y} - \frac{8x}{12y} = \frac{9x-8x}{12y} = \frac{x}{12y}
  • Simplify the resulting expression by reducing the fraction or performing any necessary arithmetic operations
  • can be used to verify the result of subtraction with unlike denominators

Simplification of complex rationals

  • After adding or subtracting rational expressions, simplify the result by the numerator and denominator, if possible
    • Look for common factors in the numerator and denominator and cancel them out
    • 6x2+3x2x−4x2−2xx=3x(2x+1)2x−2x(2x−1)x=3(2x+1)2−2(2x−1)1=6x+3−4x+22=2x+52\frac{6x^2+3x}{2x} - \frac{4x^2-2x}{x} = \frac{3x(2x+1)}{2x} - \frac{2x(2x-1)}{x} = \frac{3(2x+1)}{2} - \frac{2(2x-1)}{1} = \frac{6x+3-4x+2}{2} = \frac{2x+5}{2}
  • The simplified result is 2x+52\frac{2x+5}{2}

Techniques for rational functions

  • Apply the same techniques used for adding and subtracting rational expressions to rational functions
    • A is a function in the form of a rational expression, where both the numerator and denominator are polynomials
  • Example: Given f(x)=3x−12x+1f(x) = \frac{3x-1}{2x+1} and g(x)=2x+3x−2g(x) = \frac{2x+3}{x-2}, find (f+g)(x)(f+g)(x)
    1. (f+g)(x)=3x−12x+1+2x+3x−2(f+g)(x) = \frac{3x-1}{2x+1} + \frac{2x+3}{x-2}
    2. Find the LCD: (2x+1)(x−2)(2x+1)(x-2)
    3. Rewrite the expressions with the LCD: (3x−1)(x−2)(2x+1)(x−2)+(2x+3)(2x+1)(x−2)(2x+1)\frac{(3x-1)(x-2)}{(2x+1)(x-2)} + \frac{(2x+3)(2x+1)}{(x-2)(2x+1)}
    4. Simplify the numerators: 3x2−7x+2(2x+1)(x−2)+4x2+7x+3(x−2)(2x+1)\frac{3x^2-7x+2}{(2x+1)(x-2)} + \frac{4x^2+7x+3}{(x-2)(2x+1)}
    5. Add the numerators and keep the LCD: 3x2−7x+2+4x2+7x+3(2x+1)(x−2)=7x2+5(2x+1)(x−2)\frac{3x^2-7x+2+4x^2+7x+3}{(2x+1)(x-2)} = \frac{7x^2+5}{(2x+1)(x-2)}
  • The simplified result is (f+g)(x)=7x2+5(2x+1)(x−2)(f+g)(x) = \frac{7x^2+5}{(2x+1)(x-2)}

Domain restrictions

  • When working with rational expressions, it's crucial to consider
  • The denominator of a rational expression cannot equal zero, as division by zero is undefined
  • Identify values that make the denominator zero and exclude them from the domain of the rational expression
  • These restrictions must be considered when simplifying or combining rational expressions to ensure the final result is valid for all permissible input values

Key Terms to Review (20)

Addition of Rational Expressions: Addition of rational expressions refers to the process of combining two or more rational expressions, which are fractions with polynomials in the numerator and denominator, by finding a common denominator and then adding the numerators.
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.
Common Denominator: A common denominator is the least common multiple (LCM) of the denominators of two or more rational expressions. It is the smallest number that all the denominators can be divided into evenly, allowing for the addition and subtraction of the rational expressions.
Constant: A constant is a value or quantity that does not change within the context of a specific problem or equation. It is a fixed number or expression that remains the same throughout a mathematical operation or calculation.
Cross Multiplication: Cross multiplication is a technique used to solve proportions and rational equations. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa, to determine if the two fractions are equivalent or to solve for an unknown value.
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.
Domain Restrictions: Domain restrictions refer to the set of values for which a function or expression is defined and can be evaluated. This concept is crucial in the context of working with rational expressions and solving rational equations.
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.
Fraction Bar: The fraction bar is a horizontal line that separates the numerator and denominator in a fractional expression. It is a fundamental component of rational expressions, which are algebraic expressions that can be written in the form of a fraction.
Least Common Denominator: The least common denominator (LCD) is the smallest positive integer that is a multiple of all the denominators in a set of fractions. It is a crucial concept in working with fractions, adding and subtracting rational expressions, and solving rational equations.
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 that is particularly important in the context of adding and subtracting rational expressions, as well as adding, subtracting, and multiplying radical expressions.
Like Fractions: Like fractions are fractions that have the same denominator. They can be easily added, subtracted, multiplied, or divided because the common denominator allows for direct comparison and combination of the fractional parts.
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 Expressions: A polynomial expression is a mathematical expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers and are combined using the operations of addition, subtraction, and multiplication. Polynomial expressions are fundamental in the context of adding and subtracting rational expressions, as they form the building blocks of these operations.
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.
Simplifying: Simplifying is the process of reducing an expression to its most basic or elementary form, making it easier to understand and work with. In the context of rational expressions, simplifying involves manipulating the numerator and denominator to create an equivalent expression with the fewest possible terms or factors.
Subtraction of Rational Expressions: Subtraction of rational expressions involves finding the difference between two rational expressions, which are fractions with polynomial expressions in the numerator and denominator. This operation is essential in simplifying and manipulating rational expressions, a crucial skill in algebra.
Unlike Fractions: Unlike fractions are fractions that have different denominators, meaning the bottom numbers of the fractions are not the same. These fractions cannot be directly added or subtracted without first finding a common denominator.
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.
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