Problem 32 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Decomposition: \( \frac{1}{x} - \frac{2}{x^2} - \frac{4}{x^3} + \frac{3}{x^4} \).

Step 1: Verify Improper Fraction

Check if the degree of the numerator (x^{3}-2x^{2}-4x+3) is less than the degree of the denominator (x^4) . Since the degree of the numerator is 3 and the degree of the denominator is 4, it confirms that the fraction is proper.

Step 2: Set Up Partial Fraction Decomposition

Since the denominator is a power of a single linear term, x^{4} = x imes x imes x imes xyou can express the decomposition as a sum of terms: \[ \frac{x^{3}-2x^{2}-4x+3}{x^{4}} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4}. \]

Step 3: Clear the Denominator

Multiply through by x^4to remove the denominator: \[ x^{3} - 2x^{2} - 4x + 3 = A x^{3} + B x^{2} + C x + D. \]

Step 4: Match Coefficients

Arrange the polynomial equations by matching the coefficients from both sides:- Coefficient of x^3: \( A = 1 \) \( x^{3} \)- Coefficient of x^2: \( B = -2 \) \( x^{2} \)- Coefficient of x: \( C = -4 \) \( x \)- Constant term: \( D = 3 \).Identify the constant components (A, B, C, D) from these equations.

Key Concepts

Improper Fraction

When we're working with rational functions, we often need to verify if a given function is an *improper fraction*. A rational function is considered improper if the degree of its numerator is greater than or equal to the degree of its denominator. In this instance, the numerator is \( x^3 - 2x^2 - 4x + 3 \), which lends a degree of 3, and the denominator is \( x^4 \), yielding a degree of 4.
Since 3 is less than 4, our function \( \frac{x^3 - 2x^2 - 4x + 3}{x^4} \) is actually a *proper fraction*.
Why is this important? Well, partial fraction decomposition can only directly handle proper fractions. If we encountered an improper fraction, we would first need to perform polynomial division to change it into a proper fraction or mixed fraction form. In this case, our task is simplified because we confirmed it's already proper.

Polynomial Degree

Understanding the concept of polynomial degree is essential when dealing with terms in partial fractions. The degree of a polynomial is the highest power of the variable present. For example, in the polynomial \( x^3 - 2x^2 - 4x + 3 \), the term with the highest power is \( x^3 \), which makes 3 the degree of the polynomial.
Now, why does this matter in partial fraction decomposition? The relationship between the degree of the numerator and the denominator tells us about the nature of the rational function. If the numerator has a higher degree than the denominator, it indicates an improper fraction. In partial fraction decomposition, understanding degrees helps in allocating terms correctly for decomposing rational functions.
This knowledge helps us stay organized, especially when breaking down a fraction into simpler partial fractions, as each term will relate back to the degrees present in the function.

Coefficient Matching

Coefficient matching plays a pivotal role in the process of partial fraction decomposition. After setting up the decomposition, the next step involves equating coefficients from the polynomials involved.
Once we arrive at an expression such as \( x^3 - 2x^2 - 4x + 3 = Ax^3 + Bx^2 + Cx + D \), our task is to find the values of \(A\), \(B\), \(C\), and \(D\). Here's how it works:

• *Match the coefficient of \( x^3 \)*: yields \( A = 1 \).
• *Match the coefficient of \( x^2 \)*: gives \( B = -2 \).
• *Match the coefficient of \( x \)*: results in \( C = -4 \).
• *Match the constant term*: provides \( D = 3 \).

By equating the coefficients of corresponding powers of \( x \), we accurately determine each constant in the partial fractions, which precisely builds the breakdown of the rational function.

Rational Functions

A rational function represents a ratio of two polynomials. It is expressed in the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials and \(Q(x) eq 0\).
Rational functions are intriguing due to their diverse applications, from calculus to engineering, and are particularly significant in breaking complex functions into simpler parts through methods like partial fraction decomposition.
Understanding their behavior involves observing the numerator and denominator's degrees, forms, and factors. Most importantly in partial fraction decomposition, handling these functions with unmatched ease - whether they’re proper or improper - through logical steps enhances our ability with algebraic expression manipulations. The smooth flow from a complex rational function to individual simpler terms offers clarity and precision necessary for integration and solving equations.