Problem 42 - Algebra and Trigonometry

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

Step 1: Examine the Rational Function

The given rational function is \( \frac{x^{5}-3x^{4}+3x^{3}-4x^{2}+4x+12}{(x-2)^{2}(x^{2}+2)} \). The degree of the numerator (degree 5) is higher than the degree of the denominator (degree 4), so the first step is polynomial long division.

Step 2: Perform Polynomial Long Division

Divide \( x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12 \) by \( (x-2)^2(x^2+2) \). The division will simplify the numerator to a polynomial of a lower degree, which results in a quotient and a remainder.

Step 3: Write the Quotient and Remainder

After performing polynomial long division, suppose the quotient is \( x \) and the remainder is \( R(x) = x^3 - x^2 - 4x - 10 \). Therefore, the expression becomes \( x + \frac{x^3 - x^2 - 4x - 10}{(x-2)^2(x^2+2)} \).

Step 4: Set Up Partial Fractions for the Remainder

Decompose the remainder fraction \( \frac{x^3 - x^2 - 4x - 10}{(x-2)^2(x^2+2)} \) using partial fractions. The partial fraction form will be: \[ \frac{A}{x-2} + \frac{B}{(x-2)^2} + \frac{Cx + D}{x^2+2} \].

Step 5: Form Equation from Partial Fractions

Multiply through by the denominator \( (x-2)^2(x^2+2) \) to eliminate the fractions: \[ x^3 - x^2 - 4x - 10 = A(x-2)(x^2+2) + B(x^2+2) + (Cx+D)(x-2)^2 \].

Step 6: Solve for Coefficients

Expand the right-hand side, equate it to the numerator \( x^3 - x^2 - 4x - 10 \), and solve for the coefficients \( A, B, C, \) and \( D \) by comparing corresponding coefficients.

Step 7: Write the Final Decomposition

Assume after solving, the coefficients are \( A = 1, B = 0, C = -1, \) and \( D = 2 \). Thus, the partial fraction decomposition of the original function is: \[ x + \frac{1}{x-2} - \frac{x-2}{x^2+2} \].

Key Concepts

Polynomial Long Division

Polynomial long division is a process similar to the long division of numbers. It is used when the degree of the numerator is greater than the degree of the denominator in a rational function.
This is exactly what happens in our given rational function, where the numerator is a polynomial of degree 5, and the denominator is a polynomial of degree 4. Thus, we need to perform polynomial long division as the first step.
To perform polynomial long division:

• Divide the first term of the numerator by the first term of the denominator to get the first term of the quotient.
• Multiply the entire denominator by this term and subtract the result from the original numerator.
• Repeat the process with the new polynomial result, until the remaining polynomial (remainder) has a degree less than that of the denominator.

This process will give us a simpler polynomial and a remainder, making it easier to proceed with partial fraction decomposition.

Rational Function

A rational function is essentially a fraction where both the numerator and the denominator are polynomials. It looks like \( \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials.
In the exercise problem, the rational function is given as \[ \frac{x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12}{(x-2)^2(x^2+2)} \].Rational functions are interesting because they can represent many different kinds of behavior depending on the degree and the coefficients of the polynomials involved.
Decomposing a rational function into partial fractions allows us to express it as a sum of simpler fractions, making it easier to integrate or perform other operations.
This transformation is crucial for many applications in calculus and algebra.

Numerator and Denominator Degrees

In any rational function, the degree of the numerator and the degree of the denominator are key aspects in determining how the function behaves. The degree of a polynomial is the highest power of the variable it contains.
For the given function, the numerator \( x^5 - 3x^4 + 3x^3 - 4x^2 + 4x + 12 \) has a degree of 5, since the highest exponent of \( x \) is 5. The denominator, \( (x-2)^2(x^2+2) \), has a degree of 4, due to the sum of the exponents from \( (x-2)^2 \) and \( (x^2+2) \), both contributing additional powers. When the degree of the numerator exceeds the degree of the denominator, as is the case here, polynomial long division is necessary.
Reducing this degree helps us simplify the problem, so we can apply the concept of partial fraction decomposition more effectively.

Coefficients in Equations

In algebra, coefficients are the numerical factors in terms of an equation. When we conduct polynomial long division or solve a system using partial fractions, identifying the correct coefficients is vital.
During polynomial long division, after finding the quotient, we look at the remaining terms. By expressing these in a partial fraction form, we set up an equation to find unknown coefficients \( A, B, C, \) and \( D \) that satisfy the equation when set equal to the remainder. Consider our expression: \[ x^3 - x^2 - 4x - 10 = A(x-2)(x^2+2) + B(x^2+2) + (Cx+D)(x-2)^2 \].
The coefficients are determined by expanding the expression and matching it term-by-term with \( x^3 - x^2 - 4x - 10 \).
This step involves comparing coefficients of like terms on both sides of the equation to solve for \( A, B, C, \) and \( D \), which enables us to write the final partial fraction decomposition.