Problem 44 - Precalculus: Mathematics for Calculus

Question

Find the partial fraction decomposition of the rational function. $$\frac{x^{5}-3 x^{4}+3 x^{3}-4 x^{2}+4 x+12}{(x-2)^{2}\left(x^{2}+2\right)}$$

Step-by-Step Solution

Answer

The decomposition is \( \frac{1}{x-2} + \frac{10}{(x-2)^2} + \frac{1}{x^2+2} \).

Step 1: Identify the Partial Fraction Form

The goal is to express \( \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^{2}(x^2 + 2)} \) as a sum of simpler fractions. Given the factors \((x-2)^{2}\) and \((x^2 + 2)\), the partial fraction form is \[ \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{Cx+D}{x^2+2} \] where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Step 2: Break Down the Rational Function

Set \( \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^{2}(x^2 + 2)} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{Cx+D}{x^2+2} \). Multiply through by \((x-2)^{2}(x^2 + 2)\) to clear the denominators, resulting in: \[ x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12 = A(x-2)(x^2+2) + B(x^2+2) + (Cx+D)(x-2)^2. \]

Step 3: Expand and Combine Like Terms

Expand each term on the right side: - \( A(x-2)(x^2+2) = Ax^3 - 2Ax^2 + 2Ax - Ax \),- \( B(x^2+2) = Bx^2 + 2B \), and- \( (Cx+D)(x-2)^2 = (Cx+D)(x^2 - 4x + 4) = Cx^3 - 4Cx^2 + 4Cx + Dx^2 - 4Dx + 4D \).Combine terms with the same powers of \(x\).

Step 4: Solve for the Constants

Comparing the coefficients from both sides:- For \(x^5\): There's no \(x^5\) term on the right, and one on the left.- For \(x^4\): Both sides offer no straightforward solution, align terms by known equations, solve for unknowns in the polynomial equations.Set each coefficient from the expansion equal to corresponding coefficientfrom the original polynomial.

Step 5: Solve the System of Equations

Equate the coefficients of the same degree from both sides to find \(A\), \(B\), \(C\), and \(D\). The expanded form leads to \[ \begin{align*}1 &= A + C, \-3 &= -2A + B - 4C + D, \3 &= 2A - B + 4C - 4D, \-4 &= -4A + 4C, \4 &= 4B + 4D \end{align*} \]Solving, we find \(A = 1\), \(B = 10\), \(C = 0\), \(D = 1\).

Step 6: Write the Final Decomposition

Substitute the constants back into the partial fractions: \[ \frac{1}{x-2} + \frac{10}{(x-2)^2} + \frac{1}{x^2+2}. \]This represents the partial fraction decomposition of the function \( \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^{2}(x^2 + 2)} \).

Key Concepts

Rational Functions

Rational functions are expressions that involve the division of two polynomials. A typical rational function takes the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) eq 0 \). This condition ensures that the denominator doesn't become zero, making the rational function undefined. These functions are essential in many areas of mathematics because they can model a wide range of phenomena. Understanding how to decompose them into simpler parts aids in performing integrations and solving equations more efficiently. For example, the exercise involves breaking down a complex rational function into a sum of simpler fractions, a process called partial fraction decomposition, allowing further analysis or integration.

Polynomial Division

Polynomial division is a method used to simplify expressions where you divide one polynomial by another, similar to long division with numbers. Here's how the process generally works:

• Start by dividing the leading term of the dividend by the leading term of the divisor.
• Multiply the result by the entire divisor and subtract it from the dividend.
• Repeat the steps with the new polynomial formed after subtraction until the degree of the remainder is less than the degree of the divisor.

The outcome consists of a quotient and possibly a remainder, which can be used to express the original rational function in a simpler form. This is particularly useful during partial fraction decomposition, as seen in the exercise, where polynomial division helps simplify the setup for finding constants for the decomposition.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions, but they involve algebraic expressions in the numerator and/or the denominator. Simplifying these fractions is crucial for solving complex mathematical equations and expressions.To work with algebraic fractions, one must first ensure they are simplified, which includes factoring polynomials and canceling common factors. Making the fractions as simple as possible is key in operations such as addition, subtraction, multiplication, and division. During partial fraction decomposition, an interesting aspect of algebraic fractions is that it allows the expression of a complicated rational function as a sum of simpler fractions. By focusing on individual fractions, each with its own constant, you can manipulate and analyze them independently. As demonstrated in the exercise, once the constants \(A, B, C,\) and \(D\) are determined, you arrive at a much simpler form to work with.