Question

Decompose \(\left[\left(x^{3}+5 x^{2}+2 x-4\right) /\left[x\left(x^{2}+4\right)^{2}\right.\right.\) into partial fractions.

Short answer

The partial fractions decomposition of the given rational function is: \[F(x)=\frac{-1}{4x}+\frac{x}{4(x^2+4)}+\frac{3x+10}{(x^2+4)^2}\]

Step-by-step solution

Identify the form of partial fractions

First, we want to identify the form of the partial fractions decomposition for the given expression. Let's denote the given expression as \(F(x)\). We can write \[F(x)=\frac{x^3 +5x^2+2x-4}{x(x^2+4)^2}\] The denominator has a linear term, \(x\), and a quadratic term, \((x^2+4)\), raised to the power of 2. Therefore, the partial fractions decomposition will be of the form: \[F(x)=\frac{A}{x}+\frac{Bx + C}{x^2+4}+\frac{Dx + E}{(x^2+4)^2}\]

Clear the denominator and expand

Now, to find the coefficients A, B, C, D, and E, we multiply both sides of the equation by the denominator \(x(x^2+4)^2\): \[x^3 + 5x^2 + 2x - 4 = A(x^2+4)^2+(Bx+C)x(x^2+4)+(Dx+E)x\] Expanding the expression, we get: \[x^3 + 5x^2 + 2x - 4 = Ax^5 + 8Ax^3 + 16A + Bx^5 + Cx^4 + 4Bx^3 + 4Cx^2 + Dx^3 + Ex^2\]

Equate coefficients and solve the system of equations

Equate the coefficients of the same-degree terms and set up a system of equations to solve for A, B, C, D, and E: \[A+B=0\] (for \(x^5\)) \[C=0\] (for \(x^4\)) \[8A+4B+D=5\] (for \(x^3\)) \[16A+4C+E=2\] (for \(x^2\)) \[16A= -4\] (for \(x^0\)) Now we can find the coefficients easily by solving this system of equations: \[C=0\] From the first equation, \[B=-A\] From the last equation, \[16A=-4 \Rightarrow A=-\frac{1}{4}\] Therefore, \[B=\frac{1}{4}\] Substituting A and B into the third equation, \[8\left(-\frac{1}{4}\right)+4\left(\frac{1}{4}\right)+D=5 \Rightarrow D=3\] Substituting A and C into the fourth equation, \[16\left(-\frac{1}{4}\right)+4(0)+E=2 \Rightarrow E=10\] So, A=-1/4, B=1/4, C=0, D=3, and E=10.

Write the decomposition with the found coefficients

Now that we have found the coefficients, we can substitute them back into the decomposition form to get the final answer: \[F(x)=\frac{-\frac{1}{4}}{x}+\frac{\frac{1}{4}x + 0}{x^2+4}+\frac{3x + 10}{(x^2+4)^2}\] Simplifying, we get the partial fractions decomposition of the given rational function: \[F(x)=\frac{-1}{4x}+\frac{x}{4(x^2+4)}+\frac{3x+10}{(x^2+4)^2}\]

Key concepts

Rational Functions

Rational functions are expressions that represent the division of two polynomials. In simple terms, a rational function looks like a fraction, where both the numerator and the denominator are polynomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. For example, in the exercise you are working with, the rational function is given by:\[F(x) = \frac{x^3 + 5x^2 + 2x - 4}{x(x^2+4)^2}\]Rational functions can often have asymptotes, which are lines that the graph of the function approaches but never actually touches. The goal of working with partial fractions is often to break down the rational function into simpler fractions, allowing for easier integration and analysis. This helps in understanding behaviors of the function such as limits, roots, and surface plots.

Polynomial Decomposition

Polynomial decomposition is a technique used to break down a complex polynomial expression into simpler components. This technique is key in finding the partial fraction representation of rational functions. The idea is to express the given polynomial as a sum or product of more manageable polynomials. In the context of rational functions, polynomial decomposition involves finding polynomials in the numerator and denominator that simplify the function.

• The numerator polynomial is often taken as it is, while the denominator is examined for factorization possibilities.
• In the original exercise, the denominator polynomial \(x(x^2+4)^2\) is decomposed into different terms based on its linear and quadratic factors.

The decomposition helps develop a format that is more suited for partial fraction decomposition, easing the process of solving rational functions.

Algebraic Fractions

Algebraic fractions represent the division of algebraic expressions. These expressions can include variables, constants, and the common arithmetic operations of addition, subtraction, multiplication, and division.When dealing with algebraic fractions, particularly in rational functions, our goal is often to simplify or decompose these fractions into simpler parts. For instance, when performing partial fraction decomposition, we start with an algebraic fraction and try to express it as the sum of simpler fractions.Consider the original fraction in the exercise:\[\frac{x^3 + 5x^2 + 2x - 4}{x(x^2+4)^2}\]The goal is to break it down into simpler fractions:

• \(\frac{A}{x}\)
• \(\frac{Bx + C}{x^2 + 4}\)
• \(\frac{Dx + E}{(x^2 + 4)^2}\)

These simpler algebraic fractions make it easier to perform operations such as integration or applying particular algebraic transformations.

Coefficient Comparison

Coefficient comparison is a method used in algebra to find unknown coefficients when decomposing polynomials into simpler algebraic fractions. After setting up the expression for partial fractions, each term in the polynomial is equated by comparing coefficients of the same degree. For instance, in the given exercise, the terms of equal powers in the numerator on both sides of the equation are set equal to form a system of equations:

• For the term with \(x^5\), equate to \(A + B = 0\).
• For the term with \(x^4\), equate to \(C = 0\).
• For the term with \(x^3\), equate to \(8A + 4B + D = 5\).
• For the term with \(x^2\), equate to \(16A + 4C + E = 2\).
• For the constant term, equate to \(16A = -4\).

By solving these equations, the values of coefficients A, B, C, D, and E can be determined. This step-by-step method allows you to work through each degree of the polynomial separately, finding each coefficient one at a time and thereby confirming the decomposition is correct.