Problem 32 - Algebra and Trigonometry

Question

\(13-44=\) 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

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

Step 1: Recognize the need for partial fraction decomposition

The given function is \(\frac{x^{3}-2 x^{2}-4 x+3}{x^{4}}\). The numerator has a lower degree than the denominator, which indicates a proper rational function suitable for partial fraction decomposition.

Step 2: Set up the form of the decomposition

Identify that \(x^{4}\) in the denominator is a power of a single factor. The decomposition is based on the powers of \(x\). The form should be \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4}\).

Step 3: Multiply to clear the denominators

Multiply both sides by \(x^{4}\) to eliminate the denominators. This gives the equation \(x^3 - 2x^2 - 4x + 3 = A x^3 + B x^2 + C x + D\).

Step 4: Determine coefficients through comparison

Since we have \(x^3 - 2x^2 - 4x + 3 = Ax^3 + Bx^2 + Cx + D\), equate coefficients of corresponding powers of \(x\) from both sides:- Coefficient of \(x^3\): \(1 = A\)- Coefficient of \(x^2\): \(-2 = B\)- Coefficient of \(x\): \(-4 = C\)- Constant term: \(3 = D\)

Step 5: Solve for the coefficients

Solve the equations from Step 4:- From \(1 = A\), we have \(A = 1\).- From \(-2 = B\), we have \(B = -2\).- From \(-4 = C\), we have \(C = -4\).- From \(3 = D\), we have \(D = 3\).

Step 6: Write the partial fraction decomposition

Substitute the values of \(A, B, C,\) and \(D\) back into the form from Step 2. The partial fraction decomposition is:\[\frac{1}{x} + \frac{-2}{x^2} + \frac{-4}{x^3} + \frac{3}{x^4}\]

Key Concepts

Proper Rational Function

A proper rational function is a key concept in understanding rational functions and their decomposition. It refers to fractions where the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
For example, in the given exercise, the function \( \frac{x^{3}-2x^{2}-4x+3}{x^{4}} \) is a proper rational function. This is because the highest power of \(x\) in the numerator is 3, while in the denominator, it is 4. This difference of degrees is what makes it proper.
Proper rational functions are often simpler to decompose because the numerator is naturally smaller, simplifying the process of breaking them into simpler fractions.

Powers of x

When performing a partial fraction decomposition, understanding the powers of \(x\) is essential. This knowledge helps in setting up the structure of decomposition.
In the exercise, the denominator \(x^4\) indicates that we need to consider all the positive powers of \(x\) up to 4. This helps us to set up our partial fraction as \( \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x^4} \).
Each of these fractions corresponds to a power of \(x\) in the original denominator, ensuring the decomposition covers all terms systematically. Recognizing powers ensures that each possible divisor is accounted for in the decomposition.

Coefficients

Finding coefficients is a critical step in partial fraction decomposition. Once the fraction has been decomposed into terms involving different powers of \(x\), you need to determine the constants (coefficients) that make up each term.
For the given function, after setting the structure of the decomposition, we multiply both sides by \(x^4\) to clear the fractions, resulting in \(x^3 - 2x^2 - 4x + 3 = Ax^3 + Bx^2 + Cx + D\).
By comparing coefficients on both sides—such as those of \(x^3\), \(x^2\), \(x\), and the constant term—we can derive equations for \(A\), \(B\), \(C\), and \(D\). Solving these provides us with the correct coefficients, which are crucial for expressing the function correctly in its decomposed form.

Rational Function

A rational function is a function that can be expressed as the quotient of two polynomials. These functions are central in algebra and calculus and come in different forms depending on the degree of their numerator and denominator polynomials.
In this exercise, \( \frac{x^{3}-2x^{2}-4x+3}{x^{4}} \) is a rational function where both the nominator and denominator are polynomials, making it suitable for techniques like partial fraction decomposition.
Rational functions are essential for understanding behavioral limits, continuity, and asymptotes in graphs. Decomposing rational functions into simpler fractions makes many calculus operations, like integration, more straightforward.