Problem 31 - Precalculus: Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{4 x^{2}-x-2}{x^{4}+2 x^{3}}$$

Step-by-Step Solution

Answer

The partial fraction decomposition is \(\frac{2}{x} - \frac{1}{x^3} - \frac{2}{x+2}\).

Step 1: Factor the Denominator

To find the partial fraction decomposition, we first need to factor the denominator of the rational function. The denominator given is \(x^4 + 2x^3\). We can factor out the greatest common factor, which is \(x^3\), thus:\[x^4 + 2x^3 = x^3(x + 2)\]

Step 2: Set Up the Partial Fractions

Given the factored form of the denominator, the partial fraction decomposition will take the form:\[\frac{4x^2 - x - 2}{x^3(x + 2)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x+2}\]Here, \(A\), \(B\), \(C\), and \(D\) are constants that we need to solve for.

Step 3: Clear the Denominator

Multiply both sides by the common denominator \(x^3(x + 2)\) to clear the fractions:\[4x^2 - x - 2 = A(x^2(x + 2)) + B(x(x + 2)) + C(x + 2) + D(x^3)\]

Step 4: Expand and Collect Like Terms

Expand the right-hand side and collect like terms:\[= Ax^3 + 2Ax^2 + Bx^2 + 2Bx + Cx + 2C + Dx^3\]Combine like terms:\[= (A + D)x^3 + (2A + B)x^2 + (2B + C)x + 2C\]

Step 5: Set Up the System of Equations

By equating coefficients for each power of \(x\), we get:\[\begin{align*}A + D & = 0 \2A + B & = 4 \2B + C & = -1 \2C & = -2\end{align*}\]

Step 6: Solve the System of Equations

Starting with the simplest equation, solve for \(C\):\[2C = -2 \quad \Rightarrow \quad C = -1\]Substitute \(C = -1\) into \(2B + C = -1\):\[2B - 1 = -1 \quad \Rightarrow \quad 2B = 0 \quad \Rightarrow \quad B = 0\]Substitute \(B = 0\) into \(2A + B = 4\):\[2A = 4 \quad \Rightarrow \quad A = 2\]Finally, substitute \(A = 2\) into \(A + D = 0\):\[2 + D = 0 \quad \Rightarrow \quad D = -2\]

Step 7: Write the Partial Fractions

Now that we have the values of the constants, we can write the partial fraction decomposition:\[\frac{4x^2 - x - 2}{x^3(x + 2)} = \frac{2}{x} + \frac{0}{x^2} + \frac{-1}{x^3} + \frac{-2}{x+2}\]Simplifying:\[= \frac{2}{x} - \frac{1}{x^3} - \frac{2}{x+2}\]

Key Concepts

Rational Function

A rational function is a type of function that is expressed as the fraction of two polynomials. Specifically, it has the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not equal to zero. In the context of partial fraction decomposition, rational functions are particularly significant since the process involves breaking down a complex rational expression into simpler fractions.

This decomposition is immensely helpful in various fields such as calculus, where integrating complex rational functions becomes more manageable when they are expressed in simpler forms. A classic example would be rewriting the rational function \(\frac{4x^2 - x - 2}{x^3(x + 2)}\) as a sum of simpler fractions like \(\frac{2}{x} - \frac{1}{x^3} - \frac{2}{x+2}\). This simplification aids not only in integration but also in comprehending the behavior of the function as the variable approaches certain values, particularly values where the denominator is zero, leading to asymptotes or discontinuities.

Factoring Polynomials

Factoring polynomials is a core skill in algebra that allows us to express a polynomial as a product of its simpler components. It is essential in tackling rational functions, especially when applying partial fraction decomposition. The process of factoring simplifies complex expressions and helps identify the roots of the polynomial.

In our exercise, we started with the polynomial in the denominator \(x^4 + 2x^3\). Factoring this expression requires identifying the greatest common factor (GCF). Here, the GCF is \(x^3\), allowing the polynomial to be rewritten as \(x^3(x + 2)\). By factoring out \(x^3\), we obtain two simpler components: \(x^3\) and \(x + 2\).

These factors are indispensable in setting up the partial fraction decomposition because they determine the form of the decomposed fractions. Each factor corresponds to a potential separate fraction in the decomposition, such as \(\frac{A}{x}\), \(\frac{B}{x^2}\), and so on. This systematic breakdown allows us to solve for unknown constants, which are critical in successfully decomposing the rational function.

System of Equations

A system of equations is a collection of equations that share the same set of variables. In the context of partial fraction decomposition, solving a system of equations is a crucial step in determining the unknown constants within the fractions. When our rational expression is decomposed, like in the example \(\frac{4x^2 - x - 2}{x^3(x + 2)}\), we arrive at a series of equations that need to be simultaneously satisfied.

In the given solution, after expanding and rearranging terms, we equated coefficients for different powers of \(x\) to form a system of equations, such as:

• \(A + D = 0\)
• \(2A + B = 4\)
• \(2B + C = -1\)
• \(2C = -2\)

Solving these equations step-by-step helps us find the values of \(A\), \(B\), \(C\), and \(D\). This systematic approach ensures each part of the decomposed fraction accurately represents part of the original function. A deep understanding of solving systems of equations can greatly simplify the process and ensure accuracy in the decomposition.