11.4 Partial Fractions
2 min read•june 24, 2024
is a powerful tool for simplifying complex . It breaks down tricky fractions into simpler ones, making them easier to work with in algebra and calculus.
This technique is crucial for integrating rational functions and solving equations. By denominators and setting up partial fractions, you can tackle problems that once seemed impossible to solve.
Partial Fraction Decomposition
Decomposition of rational expressions
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- Breaks down complex rational expressions into simpler fractions
- Useful for simplifying expressions, integrating rational functions, and solving equations
- Involves the denominator and setting up partial fractions for each factor
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- :
- :
- Solve for constants by multiplying by the LCD or substituting values ()
Breakdown of complex rational expressions
- Factor the denominator completely to identify the types of factors
- Set up partial fractions based on the factor types
- Each non-repeated linear factor gets a constant numerator ()
- Repeated get increasing powers in the numerator ()
- Irreducible get a linear numerator ()
- Solve for the constants using the LCD method or substitution
- Write the simplified expression as the sum of the partial fractions
Techniques for partial fraction decomposition
- Factor the denominator completely
- Set up partial fractions for each factor type
- Multiply both sides by the LCD or substitute values to create equations
- Solve the system of equations for the constants
- Write the final answer as the sum of the partial fractions
- LCD method: Multiply both sides by the LCD and equate coefficients (using )
- : Substitute convenient values for () to create equations
Preparatory Steps for Partial Fractions
- : Used when the degree of the numerator is greater than or equal to the degree of the denominator
- : An efficient method for dividing polynomials by linear factors
- : Used to factor quadratic expressions that cannot be factored using real numbers
Applications of partial fractions
- Simplifying complex rational expressions before integration or equation solving
- Solving systems of equations involving rational expressions
- Express each equation as a rational expression
- Use to break down the expressions
- Solve the resulting system using substitution, elimination, or matrices
- Interpret the solution in the context of the original problem
- Partial fractions make complex problems more manageable by breaking them into simpler components
Key Terms to Review (24)
A/(x-a): A/(x-a) is a type of rational expression, where A represents a constant and a represents a value that the variable x is subtracted from. This expression is commonly encountered in the context of partial fractions, a technique used to decompose a rational expression into a sum of simpler rational expressions.
Ax+B/(ax^2+bx+c): Ax+B/(ax^2+bx+c) is a rational expression that can be used to represent a function in the context of partial fractions. It consists of a linear term (Ax+B) divided by a quadratic expression (ax^2+bx+c), where A, B, a, b, and c are constants.
Complex Conjugates: Complex conjugates are a pair of complex numbers that have the same real part, but their imaginary parts have opposite signs. This relationship between complex numbers is important in the study of polynomial functions and partial fractions.
Cover-up Method: The cover-up method is a technique used to solve partial fraction decomposition problems. It involves systematically canceling out factors in the denominator of a rational expression to isolate and identify the individual partial fraction terms.
Cross Multiplication: Cross multiplication is a technique used to solve for an unknown variable in a proportion or rational equation. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and then setting the two resulting products equal to each other.
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.
Improper Fraction: An improper fraction is a fractional representation where the numerator is greater than the denominator. This means the value of the fraction is greater than one whole unit.
Irreducible Quadratic Factors: Irreducible quadratic factors are polynomial expressions of degree two that cannot be further factored into smaller polynomial factors. These factors play a crucial role in the process of partial fractions, which is a technique used to decompose rational expressions into simpler, more manageable forms.
Least Common Denominator (LCD): 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 mathematics, particularly in the context of operations involving fractions, such as adding, subtracting, multiplying, and dividing.
Linear Factors: Linear factors are the linear expressions that, when multiplied together, make up a polynomial expression. They represent the individual terms that contribute to the overall polynomial function.
Non-Repeated Linear Factors: Non-repeated linear factors refer to the unique linear factors in the denominator of a rational expression that appear only once. These factors are crucial in the process of partial fraction decomposition, which is a technique used to express a rational expression as a sum of simpler rational expressions.
Partial fraction decomposition: Partial fraction decomposition is a method used to express a rational function as the sum of simpler fractions. This technique is particularly useful for integrating rational functions.
Partial Fraction Decomposition: Partial fraction decomposition is a technique used in calculus and algebra to express a rational function as a sum of simpler rational functions. It is a fundamental tool in the study of integration and the analysis of rational expressions.
Polynomial Division: Polynomial division is the process of dividing one polynomial by another to find the quotient and remainder. It is a fundamental operation in algebra that allows for the factorization and simplification of polynomial expressions.
Polynomial Long Division: Polynomial long division is a method used to divide one polynomial by another polynomial. It involves a step-by-step process of dividing the terms of the dividend by the terms of the divisor, similar to the long division algorithm used for dividing integers.
Proper Fraction: A proper fraction is a fraction where the numerator is less than the denominator, resulting in a value that is less than 1. Proper fractions are an important concept in the context of partial fractions, as they represent the fractional components of a rational expression that can be broken down into simpler terms.
Quadratic Factors: Quadratic factors refer to the factors of a quadratic expression, which is a polynomial equation of the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. Quadratic factors are used to simplify and solve quadratic expressions, which are important in the context of partial fractions.
Rational Expressions: A rational expression is a mathematical expression that can be written as a ratio of two polynomial functions. It represents the quotient of two algebraic expressions, where the denominator is never zero.
Repeated Linear Factors: Repeated linear factors refer to the occurrence of linear factors, such as $(x-a)$, that appear more than once in the denominator of a rational expression. These repeated factors are crucial in the context of partial fraction decomposition, as they require a specific approach to properly handle them.
Substitution method: The substitution method is a technique for solving systems of equations by substituting one equation into another. This transforms the system into a single-variable equation that can be solved more easily.
Substitution Method: The substitution method is a technique used to solve systems of linear equations, systems of nonlinear equations, and other types of equations by substituting one variable in terms of another. This method involves isolating a variable in one equation and then substituting that expression into the other equation(s) to solve for the remaining variable(s).
Synthetic division: Synthetic division is a simplified method of dividing polynomials where only the coefficients are used. It is particularly useful for dividing by linear factors of the form $x - c$.
Synthetic Division: Synthetic division is a shortcut method used to divide a polynomial by a linear expression of the form $(x - a)$. It allows for the efficient computation of polynomial division, providing a streamlined approach to determining the quotient and remainder of the division process.