Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{x^{3}}{x^{3}-3 x^{2}+9 x-27} $$

Step-by-Step Solution

Answer

The partial fraction decomposition is \( \frac{3/2}{x-3} - \frac{1/2}{x} - \frac{3/2}{x^2+9} \).

Step 1: Define the Polynomial

Identify that the given expression is a rational function \( \frac{x^3}{x^3 - 3x^2 + 9x - 27} \) and note that both the numerator and denominator are polynomials of degree 3.

Step 2: Factor the Denominator

Recognize that the denominator \( x^3 - 3x^2 + 9x - 27 \) is a cubic polynomial. Begin by factoring it using synthetic division or by recognizing a pattern. Notice it can be factored as \((x-3)(x^2 + 0x + 9)\).

Step 3: Verify Completeness of Factorization

Check if \(x^2 + 0x + 9\) can be factored further. It cannot be factored into real linear factors since it has no real roots (discriminant is negative \(-4 \times 9 < 0\)). So, the factorization of the denominator is complete as \((x-3)(x^2 + 9)\).

Step 4: Set Up Partial Fractions

Since the denominator is now factored into \((x-3)(x^2 + 9)\), the partial fraction decomposition will be of the form \( \frac{A}{x-3} + \frac{Bx + C}{x^2 + 9} \).

Step 5: Solve for Coefficients

Multiply both sides by the common denominator \((x-3)(x^2 + 9)\), giving \[ x^3 = A(x^2 + 9) + (Bx + C)(x-3) \].Expand and collect like terms: \( Ax^2 + 9A + Bx^2 - 3Bx + Cx - 3C \).Combine to match coefficients: \[ (A + B)x^2 + (C - 3B)x + (9A - 3C) = x^3 \].

Step 6: Determine Coefficients by Matching Terms

Equate coefficients from both sides of the equation: 1. For \(x^3\) term: Coefficient is 0.2. For \(x^2\) term: \(A + B = 1\).3. For \(x\) term: \(C - 3B = 0\).4. For constant term: \(9A - 3C = 0\).

Step 7: Solve Simultaneous Equations

Solve the system of equations:- \(A + B = 1\)- \(C - 3B = 0\)- \(9A - 3C = 0\)These simplify to \(C = 3B\) and \(A = \frac{1}{3}C\). Substitute values into \(A + B = 1\).Upon solving, we get:1. \(A = \frac{3}{2}\)2. \(B = -\frac{1}{2}\)3. \(C = -\frac{3}{2}\).

Step 8: Write the Partial Fraction Decomposition

Substitute back into the partial fraction form:\( \frac{x^3}{x^3 - 3x^2 + 9x - 27} = \frac{\frac{3}{2}}{x-3} - \frac{1}{2}x - \frac{3}{2}{x^2 + 9} \).

Key Concepts

Rational Functions

A rational function is simply a fraction that consists of two polynomials. The polynomial in the numerator can be of any degree, and the same goes for the polynomial in the denominator, as long as the denominator is not zero.
In the given exercise, the rational function is \( \frac{x^3}{x^3 - 3x^2 + 9x - 27} \).
It's essential to understand that the degree of the polynomial in the numerator and the denominator plays a critical role in the analysis and simplification of these functions.

• If the degree of the numerator is less than the degree of the denominator, the rational function is a proper rational function.
• If the degree of the numerator is equal to or greater than the degree of the denominator, the rational function is improper, necessitating further simplification methods like polynomial long division.

Recognizing rational functions and their characteristics is the first step toward breaking them down into partial fractions.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of simpler polynomials. This process is crucial in partial fraction decomposition because it enables you to express a complex fraction as a sum of simpler fractions. In the exercise, it's necessary to factor the cubic polynomial \( x^3 - 3x^2 + 9x - 27 \).
To do this, look for any patterns or use methods such as synthetic division or recognizing common factors.
In our case, the polynomial factors neatly into \((x-3)(x^2 + 0x + 9)\).

• Start by trying to recognize a pattern or easy factor; for example, using the difference of cubes or other common identities can help.
• If direct methods do not work, try synthetic division to simplify the polynomial into factors.

Remember, factoring is complete when you can't break it down into any more real or complex linear factors.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear factor. This technique can quickly help in factoring polynomials.
For the polynomial \( x^3 - 3x^2 + 9x - 27 \), checking for possible roots like \( x = 3 \) can streamline the process.
If a suspected root divides the polynomial without a remainder, it's confirmed as a factor.

• Write down the coefficients of the polynomial.
• Choose a suspected root, such as numbers that might logically divide the constant term.
• Perform the process step by step, bringing down, multiplying, and adding, to see if you get a remainder of zero.

Once a factor is known, use it to further factor the rest of the polynomial or confirm the factorization you've started.

System of Equations

Often in partial fraction decomposition, setting up a system of equations is necessary to determine unknown coefficients. Once the denominator is factored, these coefficients help you express the original rational function in terms of simpler fractions.
In the exercise, we used the factored form \( \frac{A}{x-3} + \frac{Bx + C}{x^2 + 9} \).
Equating and expanding both sides of the equation results in a system of equations:

• \( A + B = 1 \)
• \( C - 3B = 0 \)
• \( 9A - 3C = 0 \)

Solving these equations provides the values for \( A \), \( B \), and \( C \), which complete the decomposition of the polynomial into partial fractions. It often involves simple algebraic steps like substituting terms and simplifying expressions to find the correct coefficients. This algebraic step is vital in expressing complex rational functions in a simpler, decomposed form.