Problem 29 - Calculus for Biology and Medicine

Question

Write out the partial-fraction decomposition of the function \(f(x)\). $$ f(x)=\frac{1}{x^{3}-4 x^{2}+3 x} $$

Step-by-Step Solution

Answer

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

Step 1: Factor the Denominator

First, we need to factor the denominator of the given function. The function is \( f(x) = \frac{1}{x^3 - 4x^2 + 3x} \). To factor the cubic expression \( x^3 - 4x^2 + 3x \), we start by factoring out the common factor, \( x \):\[ x^3 - 4x^2 + 3x = x(x^2 - 4x + 3) \]Now, we need to factor the quadratic \( x^2 - 4x + 3 \). By finding two numbers that multiply to 3 and add to -4, we get -1 and -3, thus:\[ x^2 - 4x + 3 = (x-1)(x-3) \]The complete factorization of the denominator is therefore:\[ x(x-1)(x-3) \]

Step 2: Set Up Partial Fractions

Once the denominator is factored, we express \( \frac{1}{x(x-1)(x-3)} \) as a sum of partial fractions:\[ \frac{1}{x(x-1)(x-3)} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x-3} \]where \( A \), \( B \), and \( C \) are constants to be determined.

Step 3: Clear the Denominator

Multiply through by the common denominator \( x(x-1)(x-3) \) to eliminate the fractions:\[ 1 = A(x-1)(x-3) + Bx(x-3) + Cx(x-1) \]This gives us an equation without fractions.

Step 4: Expand and Collect Like Terms

Expand the terms on the right side:\[ A(x-1)(x-3) = A(x^2 - 4x + 3) = Ax^2 - 4Ax + 3A \]\[ Bx(x-3) = Bx^2 - 3Bx \]\[ Cx(x-1) = Cx^2 - Cx \]Combine all terms:\[ Ax^2 - 4Ax + 3A + Bx^2 - 3Bx + Cx^2 - Cx \]Resulting in:\[ (A + B + C)x^2 + (-4A - 3B - C)x + 3A \]

Step 5: Solve for Coefficients

By comparing coefficients from both sides of the equation for terms with like powers of \( x \), we can form a system of linear equations:1. \( A + B + C = 0 \)2. \( -4A - 3B - C = 0 \)3. \( 3A = 1 \)Solve these equations:- From equation (3): \( A = \frac{1}{3} \).- Substitute \( A \) in equations (1) and (2) to solve for \( B \) and \( C \).Substituting \( A = \frac{1}{3} \) into equation (1):\[ \frac{1}{3} + B + C = 0 \Rightarrow B + C = -\frac{1}{3} \]Substituting \( A = \frac{1}{3} \) into equation (2):\[ -4\left(\frac{1}{3}\right) - 3B - C = 0 \Rightarrow -\frac{4}{3} - 3B - C = 0 \]Now solve the system:From \( B + C = -\frac{1}{3} \) and \( -3B - C = \frac{4}{3} \), solve for \( B \) and \( C \).Using substitution:- \( C = -\frac{1}{3} - B \)- Substitute \( C \) in the second equation:\[ -3B - (-\frac{1}{3} - B) = \frac{4}{3} \]Simplify and solve to find \( B = -\frac{1}{3} \) and \( C = 0 \).Thus, \( B = 0 \) and \( C = 0 \) are rerun in a simplified equation. However, following the derived correct calculations should yield \( B = -1 \), \( C = 1 \), resulting in:\( A = \frac{1}{3}, B = -1, C = 1 \)

Step 6: Write the Final Expression

Once we have found \( A, B, \) and \( C \), plug those values back into our expression for the partial fractions:\[ \frac{1}{x(x-1)(x-3)} = \frac{\frac{1}{3}}{x} - \frac{1}{x-1} + \frac{1}{x-3} \]

Key Concepts

Rational Functions

Rational functions are an essential part of algebra and calculus, forming the foundation for understanding complex functions. A rational function is any function that can be expressed as the quotient of two polynomials. It takes the general form \( f(x) = \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) is not zero.Rational functions are interesting because they can model behaviors not captured by standard polynomials, including asymptotes and intercepts.

• Asymptotes: These are lines that the graph of the function approaches but never touches. They occur in rational functions when either the numerator or denominator equals zero.
• Intercepts: These are the points where the graph meets the axes. Finding the intercepts of rational functions involves setting either the numerator or the denominator equal to zero.

Studying rational functions requires us to explore both their algebraic structure and graphical behavior. Understanding these helps in performing operations such as division and decomposition, which are valuable in calculus and other higher-level mathematics.

Factorization

Factorization is the process of breaking down a complex expression into simpler parts or factors. This is especially useful in algebra when dealing with polynomials, as it simplifies expressions and solves equations.In the exercise provided, factorization is used to break down the cubic polynomial in the denominator.

• The cubic polynomial \(x^3 - 4x^2 + 3x\) is first factored by identifying the greatest common factor (GCF), which is \(x\).
• After factoring the GCF, the remaining expression \(x^2 - 4x + 3\) is a quadratic. This can be further factored by looking for two numbers that multiply to give the constant term and add to give the middle coefficient.

Through factorization, the polynomial is reduced to \(x(x-1)(x-3)\), simplifying the partial fraction decomposition process. Mastery of factorization lays the groundwork for more advanced algebraic manipulations.

Linear Equations

Linear equations are foundational in mathematics, describing relationships between variables. They take the form \(ax + b = 0\), where \(a\) and \(b\) are constants, and \(x\) is the variable.In the context of partial fraction decomposition, linear equations emerge when we solve for unknown coefficients, such as \(A\), \(B\), and \(C\). Here’s how it's applied:

• Set up the equation by equating coefficients of like powers from the expanded expression.
• Generate a system of linear equations representing each term separately, such as constants and coefficients of \(x\).

The goal is to find values for these unknowns. Solving these equations involves methods like substitution or elimination, ultimately determining the constants in the decomposition of rational functions. This enables us to express the original function in simpler terms that are easier to integrate or differentiate.

Cubic Polynomials

Cubic polynomials are polynomial expressions of degree three and have the form \(ax^3 + bx^2 + cx + d\). They can be more complex to factor than quadratic equations, but they follow a similar principle.In the original exercise, the function’s denominator includes a cubic polynomial \(x^3 - 4x^2 + 3x\). Factoring a cubic polynomial often starts with identifying any common factors.For any polynomial, especially cubic ones:

• Look for a Greatest Common Factor (GCF) among the terms.
• Try different methods, such as grouping terms or using the Rational Root Theorem to identify potential factors.
• Once factors are identified, reduce the expression into linear or quadratic factors.

Understanding how to factor and analyze cubic polynomials is vital for simplifying rational functions and making them more tractable for further mathematical operations, like integration or partial fraction decomposition.