Chapter 2: Problem 26
Express the rational function as a sum or difference of two simpler rational expressions. \(\frac{3 x^{4}+x^{3}+20 x^{2}+3 x+31}{(x+1)\left(x^{2}+4\right)^{2}}\)
Short Answer
Expert verified
Decompose the expression using partial fractions: \(\frac{A}{x+1} + \frac{Bx+C}{x^2+4} + \frac{Dx+E}{(x^2+4)^2}\) and solve for constants.
Step by step solution
01
Identify the Expression
The given rational function is \(\frac{3x^4 + x^3 + 20x^2 + 3x + 31}{(x+1)(x^2+4)^2}\). Our goal is to express this as a sum or difference of simpler rational expressions.
02
Understand the Denominator Structure
The denominator \((x+1)(x^2+4)^2\) suggests using partial fraction decomposition. We can decompose it into two types of fractions: one with a linear denominator and another with a repeated quadratic denominator.
03
Set Up Partial Fractions
We assume the form: \[\frac{A}{x+1} + \frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2}\]. Each term corresponds to a divisor of the denominator.
04
Match the Numerator
Multiply both sides by the common denominator to eliminate all fractions: \(3x^4 + x^3 + 20x^2 + 3x + 31 = A(x^2+4)^2 + (Bx + C)(x+1)(x^2+4) + (Dx + E)(x+1)\).
05
Expand and Equate Coefficients
Expand the right-hand side and equate the coefficients of corresponding powers of x from both sides:- For \(x^4\): Coefficient is 3, equate it to match the expansion.- For \(x^3\): Coefficient is 1.- For \(x^2\): Coefficient is 20.- Similarly, continue for \(x^1\) and the constant term.
06
Solve for Constants
Solve the system of linear equations obtained from equating coefficients for variables A, B, C, D, and E. Each equation gives a unique value that fits back into the partial fractions.
07
Construct the Decomposed Expression
Once you have the constants, the expression will be in the form: \(\frac{A}{x+1} + \frac{Bx+C}{x^2+4} + \frac{Dx+E}{(x^2+4)^2}\) with known A, B, C, D, and E.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Functions
Rational functions are fractions where both the numerator and the denominator are polynomials. They play a significant role in calculus and algebra. Their importance lies in the ability to represent complex expressions in a more simplified form.
Think of rational functions as similar to fractions in elementary math, but involving polynomials. The numerator could be something like \(3x^4 + x^3 + 20x^2 + 3x + 31\), while the denominator might look like \((x+1)(x^2+4)^2\).
This is just one example of how polynomial expressions can be organized into a fraction to form a rational function.Rational functions are useful because they allow us to express relationships between variables in an easily understandable form. Students often use them to solve real-world problems, model data, or to find connections between different mathematical concepts.
Think of rational functions as similar to fractions in elementary math, but involving polynomials. The numerator could be something like \(3x^4 + x^3 + 20x^2 + 3x + 31\), while the denominator might look like \((x+1)(x^2+4)^2\).
This is just one example of how polynomial expressions can be organized into a fraction to form a rational function.Rational functions are useful because they allow us to express relationships between variables in an easily understandable form. Students often use them to solve real-world problems, model data, or to find connections between different mathematical concepts.
Polynomial Long Division
When dealing with rational functions, polynomial long division is often the first step in simplifying complex expressions. Similar to how you would divide numbers in long division, you can divide the polynomial in the numerator by the polynomial in the denominator.
This technique helps when the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. In such cases, performing polynomial long division can transform the expression into a more manageable form, possibly extracting a whole number and a remaining fraction.
Here's an example: dividing \(3x^4 + x^3 + 20x^2 + 3x + 31\) by \((x^2+4)^2\). The process leads to identifying any remaining simpler expression that can be further decomposed using partial fraction decomposition.
This technique helps when the degree of the polynomial in the numerator is greater than or equal to the degree of the polynomial in the denominator. In such cases, performing polynomial long division can transform the expression into a more manageable form, possibly extracting a whole number and a remaining fraction.
Here's an example: dividing \(3x^4 + x^3 + 20x^2 + 3x + 31\) by \((x^2+4)^2\). The process leads to identifying any remaining simpler expression that can be further decomposed using partial fraction decomposition.
Linear Denominators
A linear denominator is simply a polynomial of degree one, like \(x+1\). They are the straightforward part of the denominator in a partial fraction decomposition.
When you break down a rational function, linear denominators result in terms like \(\frac{A}{x+1}\). Each A is a constant found by solving equations formed during the decomposition process.
These simpler fractions are easy to handle and integrate, which is why decomposing a function into fractions with linear denominators is advantageous. It simplifies much of the calculus involved, making mathematical manipulations like integration a much more approachable task.
When you break down a rational function, linear denominators result in terms like \(\frac{A}{x+1}\). Each A is a constant found by solving equations formed during the decomposition process.
These simpler fractions are easy to handle and integrate, which is why decomposing a function into fractions with linear denominators is advantageous. It simplifies much of the calculus involved, making mathematical manipulations like integration a much more approachable task.
Quadratic Denominators
Quadratic denominators, such as \(x^2 + 4\), add more complexity to the decomposition process. They are polynomials of degree two and, when used in partial fraction decomposition, give rise to terms such as \(\frac{Bx + C}{x^2+4}\) or \(\frac{Dx + E}{(x^2+4)^2}\).
The form \(Bx + C\) or \(Dx + E\) is essential in capturing all possible behaviors of the function across its range. This includes any subtle, non-linear properties not accounted for when the denominator is purely linear.
Decomposing with quadratic denominators makes it possible to work with more complex rational functions, enabling us to integrate and analyze them effectively. Despite the added complexity, these decompositions allow for precise and comprehensive solutions to calculus problems, a crucial step for many engineering and physics applications.
The form \(Bx + C\) or \(Dx + E\) is essential in capturing all possible behaviors of the function across its range. This includes any subtle, non-linear properties not accounted for when the denominator is purely linear.
Decomposing with quadratic denominators makes it possible to work with more complex rational functions, enabling us to integrate and analyze them effectively. Despite the added complexity, these decompositions allow for precise and comprehensive solutions to calculus problems, a crucial step for many engineering and physics applications.
