Problem 56 - College Algebra

Question

For the following exercises, find the partial fraction expansion. $$\frac{x^{3}-4 x^{2}+5 x+4}{(x-2)^{3}}$$

Step-by-Step Solution

Answer

\(\frac{1}{x-2} + \frac{9}{(x-2)^2} + \frac{18}{(x-2)^3}\)

Step 1: Identify the Form of Partial Fractions

Given the function \( \frac{x^{3}-4 x^{2}+5 x+4}{(x-2)^{3}} \), notice that the denominator is \((x-2)^3\). This suggests the partial fraction has the form \( \frac{A}{(x-2)} + \frac{B}{(x-2)^2} + \frac{C}{(x-2)^3} \), where \(A, B, \) and \(C\) are constants that need to be determined.

Step 2: Multiply Through by the Denominator

To clear the fractions, multiply both sides by the denominator \((x-2)^3\). This gives:\[ x^3 - 4x^2 + 5x + 4 = A(x-2)^2 + B(x-2) + C \]

Step 3: Expand and Collect Like Terms

Expand the right side:1. \(A(x-2)^2 = A(x^2 - 4x + 4)\)2. \(B(x-2) = Bx - 2B\)3. Combine all terms: \[ A(x^2 - 4x + 4) + (Bx - 2B) + C \]4. Which is \( Ax^2 - 4Ax + 4A + Bx - 2B + C \) 5. Combine like terms to get: \( Ax^2 + (B-4A)x + (4A-2B+C) \)

Step 4: Match Coefficients with the Original Polynomial

Compare the coefficients from \(Ax^2 + (B-4A)x + (4A-2B+C)\) with \(x^3 - 4x^2 + 5x + 4\). The expanded form on the left does not include an \(x^3\) term, implying the coefficient of \(x^3\) is 1, and therefore:- \( A = 1 \)The coefficients of corresponding terms are:- For \(x^2\): \(-4 = A\) - For \(x\): \(5 = B - 4A \)- For the constant: \(4 = 4A - 2B + C \)

Step 5: Solve the System of Equations

Substitute \( A = 1 \) into the equations:1. \( -4 = 1 \) (This checks out as we correctly noted no explicit \(x^2\) term)2. \( 5 = B - 4(1) \Rightarrow B = 9 \)3. \( 4 = 4(1) - 2(9) + C \Rightarrow 4 = 4 - 18 + C \Rightarrow C = 18 \)

Step 6: Write the Partial Fraction

Substitute \(A = 1\), \(B = 9\), and \(C = 18\) back into the partial fraction expression:\[ \frac{x^3 - 4x^2 + 5x + 4}{(x-2)^3} = \frac{1}{x-2} + \frac{9}{(x-2)^2} + \frac{18}{(x-2)^3} \]

Key Concepts

Polynomial Division

Understanding polynomial division is key to working with rational expressions. It is similar to dividing numbers, but involves polynomials instead. Polynomial division becomes crucial when the degree of the numerator is greater than or equal to the degree of the denominator.

There are two main types of polynomial division:

• Long division
• Synthetic division (used when the divisor is of the form x - c)

In our problem, the numerator is a cubic polynomial while the denominator is a cubic polynomial as well, meaning they both have the same degree.
Polynomial division helps ensure any possible simplifications are undertaken before attempting further decompositions. However, in this problem, the degree of the numerator and denominator doesn't require pre-dividing, enabling us to move directly to forming partial fractions.

Rational Expressions

A rational expression is a fancy term for fractions that involve polynomials. It is of the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x) eq 0\).

Rational expressions can often be simplified by factoring the numerator and denominator, and canceling any common factors. In our example:

• The numerator is \(x^{3}-4x^{2}+5x+4\)
• The denominator is \((x-2)^{3}\)

Given the structure of the denominator, we find the particular structure for partial fraction decomposition that fits our rational expression.
This process allows us to express a complex fraction as a set of simpler fractions that are easier to integrate or differentiate if needed.

System of Linear Equations

Solving a system of linear equations often appears in partial fraction decomposition. Once the partial fractions are set up, equating coefficients creates a system of equations which need to be solved to find unknown constants.

In the case of our example, after expressing the rational expression as partial fractions, three equations appeared, derived from:

• Matching the \(x^2\) terms
• Matching the \(x\) terms
• Matching the constant terms

From the system:

• Equation for \(A\) was derived from the nonexistence of the \(x^3\) term.
• The equation for \(B\) was adjusted based on the coefficients of the linear terms.
• The equation for \(C\) was found using the constant terms.

Using substitution, we find \(A = 1\), \(B = 9\), and \(C = 18\). Solving these ensures that the partial fraction decomposition accurately represents the original rational expression.