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 partial fraction decomposition is \( x - 3 + \frac{10}{x-2} + \frac{12}{(x-2)^2} - \frac{4x}{x^2+2} \).

Step 1: Confirm Properness

First, check whether the given rational function \( \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^2(x^2+2)} \) is proper. A rational function is proper if the degree of the numerator is less than the degree of the denominator. Here, the degree of the numerator is 5 and the degree of the denominator is 4. Hence, perform synthetic division on the function to make it proper.

Step 2: Perform Synthetic Division

Perform synthetic division of \( x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12 \) by \( (x-2)^2(x^2+2) = x^4 - 4x^3 + 4x^2 + 2x^2 - 8x + 4 \) to find the polynomial quotient and remainder. After performing synthetic division, we get:\[ x - 3 - \frac{10x^3 - 10x^2 - 8}{(x-2)^2(x^2+2)} \]

Step 3: Set Up Partial Fractions

Setup partial fraction decomposition for the rational part \( \frac{10x^3 - 10x^2 - 8}{(x-2)^2(x^2+2)} \). Each factor in the denominator contributes to a term in the partial fraction decomposition:\[ \frac{10x^3 - 10x^2 - 8}{(x-2)^2(x^2+2)} = \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{Cx+D}{x^2+2} \]

Step 4: Eliminate Denominator

Multiply through by the common denominator \((x-2)^2(x^2+2)\) to eliminate it:\[ 10x^3 - 10x^2 - 8 = A(x-2)(x^2+2) + B(x^2+2) + (Cx+D)(x-2)^2 \]

Step 5: Expand and Solve Coefficients

Expand the right side of the equation and equate coefficients with the left side. Solve the resulting system of equations for the unknowns \( A, B, C, \) and \( D \). After equating coefficients, we solve:\[\begin{align*}A &= 10 \ B &= 12 \C &= 0 \D &= -4\end{align*}\]

Step 6: Write Final Decomposition

Incorporate these values back into the partial fraction setup:\[ \frac{10x^3 - 10x^2 - 8}{(x-2)^2(x^2+2)} = \frac{10}{x-2} + \frac{12}{(x-2)^2} - \frac{4x}{x^2+2} \]

Step 7: Combine with Polynomial Quotient

Finally, combine this partial fraction with the polynomial quotient found from synthetic division:\[ x - 3 + \frac{10}{x-2} + \frac{12}{(x-2)^2} - \frac{4x}{x^2+2} \]

Key Concepts

Rational Functions

Rational functions are expressions that lead us into the world of polynomials and fractions. They appear often in algebra and calculus. A rational function is simply the ratio of two polynomial functions. In other words, it's like a fraction where both the top and bottom are polynomials.
For instance, let's consider the example we are talking about: \( \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^2(x^2+2)} \). This is a rational function because the numerator \( x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12 \) and the denominator \( (x-2)^2(x^2+2) \) are both polynomials.

• The numerator has a degree of 5.
• The denominator, an expanded form, shows us it also has a degree of 4. It contains quadratic and repeated linear factors.

Rational functions are super useful in both everyday problems and higher math problems, as they help us describe relationships and behaviors with equations.

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a linear factor without having to go through the long division process. It's efficient and saves time. However, it's applicable when dividing by a factor of the form \((x - c)\).
In our initial exercise, we have to make the rational function 'proper' by reducing the degree of the numerator using synthetic division. The numerator here, \( x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12 \), is divided by the constructed denominator \( (x-2)^2(x^2+2) \).

• First, we calculate the roots or zeros of the divisor, focusing on \((x-2)\) since it's easier with linear factors.
• Synthetic division efficiently reduces the polynomial by confirming and applying these roots.

The result gives us a new polynomial quotient of \( x - 3 \) and a remainder, which is further simplified.

Degree of Polynomial

The degree of a polynomial is a fundamental concept when working with polynomials and rational functions. It tells us a lot about the polynomial's behavior and sometimes its shape. The degree is basically the highest power of the variable within the polynomial.
Considering our example, the numerator is a fifth-degree polynomial because the highest power of \( x \) is \( x^5 \). Meanwhile, unraveling the denominator reveals a fourth-degree polynomial because the product of its highest factors reaches a power of \( x^4 \).

• The degree in the numerator: 5.
• The degree in the denominator: 4.

Knowing these degrees is crucial because they help us determine whether a rational function is proper or improper. This distinction further affects how we approach problems like partial fraction decomposition.

Proper Fraction

A proper fraction in the context of rational functions is a situation where the degree of the numerator is less than the degree of the denominator. It’s essential for simplifying and decomposing rational functions. Proper fractions allow us to break complex expressions into simpler, more manageable parts using techniques such as partial fraction decomposition.
In our exercise, the initial rational function \( \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^2(x^2+2)} \) is classified as improper because the numerator degree (5) exceeds the denominator degree (4). This necessitates an adjustment through division before proceeding with partial fraction decomposition.

• Improper fractions need conversion into proper fractions before decomposition.
• Making a fraction 'proper' might require synthetic division or long division.

Once proper, the rational function can smoothly be transformed into a sum of simpler fractions, paving the way for problem-solving and integration in calculus.