Problem 38 - Precalculus with Limits

Question

Writing the Partial Fraction Decomposition. Write the partial fraction decomposition of the rational expression. Check your result algebraically. $$\frac{3}{x^{4}+x}$$

Step-by-Step Solution

Answer

The solution involves splitting the given rational expression into simpler fractional components using the technique of Partial Fraction Decomposition which involves solving for the coefficients of the resulting fractions. After obtaining the decomposed fraction, the result can then be checked by adding the decomposed fractions to ensure they match the original fraction.

Step 1: Simplify the denominator

First, factorize the denominator into irreducible components: \(x^{4}+x\) can be written as \(x(x^{3}+1)\), further it becomes \(x(x+1)(x^2-x+1)\). Therefore, \( \frac{3}{x^{4}+x} \) can be written as \( \frac{3}{x(x+1)(x^{2}-x+1)} \).

Step 2: Split the fraction

Write the expression as partial fractions. Here, the expression is a proper fraction, so it can be written as the sum of simpler fractions as follows \( \frac{3}{x(x+1)(x^{2}-x+1)} = \frac{A}{x} + \frac{B}{x+1} + \frac{Cx+D}{x^{2}-x+1} \).

Step 3: Solving for coefficients

To solve for the coefficients \(A\), \(B\), \(C\) and \(D\), we multiply both sides by \(x(x+1)(x^{2}-x+1)\) to cancel out the denominators. This yields: \(3 = Ax(x+1)(x^{2}-x+1) + Bx(x+1)(x^{2}-x+1) + (Cx+D)x(x+1)\) . This polynomial equation should hold true for any value of x. We can make the calculations simpler by substituting convenient values for x, typically checking for roots of the denominator. So, we start by substituting x=0, x=-1, and another value for x which we can substitute is x=1. This will give us the values for A and B.

Step 4: Solve for C and D

To solve for C and D, we will expand the equation and combine like terms, so that the result can be written as \(3 = (A+C)x^{3} + (A-B+C+D)x^{2} + (A+B+C)x + A+D\). Then, you will compare the coefficients with the original equation which is \(3 = 0x^{3} + 0x^{2} + 0x + 3\). This will enable you to draft two more equations to solve for C and D.

Step 5: Obtaining the Partial Fraction Decomposition

Replace the coefficients \(A\), \(B\), \(C\) and \(D\) back into the equation obtained in step 2. This will be the final Partial Fraction Decomposition.

Step 6: Checking the result

To check if your result matches the original fraction, simply add up the individual fractions that make up the decomposed fraction. The sum should equal the original fraction \( \frac{3}{x^{4}+x} \).

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are common in algebra, as they allow us to represent more complex relationships between variables. In the exercise at hand, we are looking at the rational expression \( \frac{3}{x^4 + x} \). Here, the numerator is the constant 3, and the denominator is the polynomial \( x^4 + x \).

Working with rational expressions often involves simplifying them, factoring the polynomials, and sometimes performing operations such as addition, subtraction, multiplication, or division. The key is to make the expression easier to handle or to better understand its properties. This often involves factorization, which we’ll dive into next.

Algebraic Factorization

Algebraic factorization is the process of breaking down a polynomial into a product of smaller polynomials, called factors. The main goal is to simplify expressions and solve equations that involve polynomials. In our exercise, the denominator \( x^4 + x \) needs to be factorized to help us in finding its partial fraction decomposition.

First, we look for common factors in each term. Here, \( x \) is a common factor, so we factor it out:
\[ x(x^3 + 1) \].
Next, we aim to factor \( x^3 + 1 \) further. Recognizing the sum of cubes, this can be factored as \((x + 1)(x^2 - x + 1)\).

Thus, the complete factorization of the original denominator is \( x(x + 1)(x^2 - x + 1) \). This factorization is crucial for the decomposition process we will go through.

Polynomial Equations

Polynomial equations consist of variables raised to whole number powers. They are equations that can involve many terms and can be set equal to another polynomial or a constant. In the context of partial fraction decomposition, we often deal with polynomial equations to simplify rational expressions.

During the decomposition, we express our rational expression as a sum of simpler fractions:
\[ \frac{3}{x(x + 1)(x^2 - x + 1)} = \frac{A}{x} + \frac{B}{x+1} + \frac{Cx+D}{x^2-x+1} \].

This process transforms our task into solving a polynomial equation where we find values for the unknowns \(A, B, C, \) and \(D\). This requires manipulating and reorganizing the polynomial terms and using specific substitution techniques.

Coefficient Comparison

Coefficient comparison is a method used to find unknown coefficients in equations, particularly in polynomial equations. This approach involves aligning similar terms on both sides of the equation and ensuring their coefficients are equal. In the process of partial fraction decomposition, it is essential to determine the values of constants \(A, B, C,\) and \(D\) that make the equation true.

Once we set up the equation by canceling out the denominators, we have:
\[ 3 = Ax(x+1)(x^2-x+1) + Bx(x+1)(x^2-x+1) + (Cx+D)x(x+1) \].
To solve for these coefficients, we choose strategic values for \(x\), often the roots of the factors in the denominator (like 0 and -1) to simplify calculations.

Then, we compare the coefficients of like terms from the expanded polynomial equation to draft new equations to solve for each unknown. This step-by-step approach helps to methodically solve for \(C\) and \(D\), confirming the correct partial fraction decomposition.