Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-28: Find the partial fraction decomposition. $$ \frac{2 x^{4}-2 x^{3}+6 x^{2}-5 x+1}{x^{3}-x^{2}+x-1} $$

Step-by-Step Solution

Answer

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

Step 1: Check for Proper Rational Function

Verify if the degree of the numerator is less than the degree of the denominator. The degree of the numerator is 4 and the degree of the denominator is 3, so the given rational function is improper.

Step 2: Polynomial Long Division

Since the rational function is improper, perform polynomial long division to divide the numerator by the denominator. After performing the division, you should get:\[\frac{2x^4 - 2x^3 + 6x^2 - 5x + 1}{x^3 - x^2 + x - 1} = 2x + 1 + \frac{x^2 - 2x + 0}{x^3 - x^2 + x - 1}\]

Step 3: Express Remainder with New Rational Function

Focus on the remaining rational function \(\frac{x^2 - 2x}{x^3 - x^2 + x - 1}\), which is now a proper fraction.

Step 4: Factor the Denominator

Factor the denominator: \(x^3 - x^2 + x - 1\). Recognizing it as a cubic polynomial, factor it as \((x - 1)(x^2 + 1)\).

Step 5: Set Up Partial Fractions

Write the partial fraction decomposition for \(\frac{x^2 - 2x}{(x - 1)(x^2 + 1)}\) as:\[\frac{A}{x - 1} + \frac{Bx + C}{x^2 + 1}\]

Step 6: Clear the Denominator

Multiply through by the common denominator \((x - 1)(x^2 + 1)\) to clear the fractions:\[x^2 - 2x = A(x^2 + 1) + (Bx + C)(x - 1)\]

Step 7: Expand and Collect Terms

Expand the right-hand side and combine like terms:\[x^2 - 2x = Ax^2 + A + Bx^2 - Bx + Cx - C\]

Step 8: Equate Coefficients

Equate the coefficients of corresponding powers of \(x\):1. For \(x^2\): \(1 = A + B\)2. For \(x\): \(-2 = -B + C\)3. For constant term: \(0 = A - C\)

Step 9: Solve the System of Equations

Solve the system of equations:1. \(A + B = 1\)2. \(-B + C = -2\)3. \(A - C = 0\)From these, we find that \(A = 2, B = -1, C = 2\).

Step 10: Write the Partial Fraction Decomposition

Substitute \(A\), \(B\), and \(C\) into the partial fraction setup:\[\frac{x^2 - 2x}{(x - 1)(x^2 + 1)} = \frac{2}{x - 1} + \frac{-x + 2}{x^2 + 1}\]

Step 11: Combine Results

Combine the results from Steps 2 and 10 to write the full expression for the original problem:\[\frac{2x^4 - 2x^3 + 6x^2 - 5x + 1}{x^3 - x^2 + x - 1} = 2x + 1 + \frac{2}{x - 1} + \frac{-x + 2}{x^2 + 1}\]

Key Concepts

Polynomial Long Division

When tackling improper rational functions, polynomial long division comes in handy. An improper rational function is when the degree of the polynomial in the numerator is greater than or equal to that in the denominator. To simplify it, we divide the numerator by the denominator.
This works similarly to long division with numbers but involves polynomials instead.
Through the division process, we aim to find a simpler form by achieving a quotient and possibly a remainder, leading us to a partially simplified expression before breaking it down further.

Proper Rational Function

The concept of proper rational functions is easy to grasp. A rational function is deemed 'proper' when its numerator is of lower degree than its denominator. If it’s not, like in the provided exercise, it’s 'improper'.
For partial fraction decomposition, only proper rational functions can be directly decomposed, hence the necessity for polynomial long division initially.

• This step ensures that what remains is decomposable into simpler terms.
• Proper rational functions facilitate the task of partial fraction decomposition, allowing us to break them into simpler, more manageable pieces.

Equate Coefficients

Once we have expressed our remainder as a proper rational function and set up the partial fractions, the task of equating coefficients arises. This involves matching terms from both sides of the equation based on their power of x.
This step is crucial because:

• It allows us to determine the values of unknown constants that make our equation true.
• We solve the system of equations that arise from equating the coefficients to ensure the equality holds for all values of x.

By methodically equating and solving for these constants, we succeed in fine-tuning our partial fractions to precisely match the original expression's structure.

Factor the Denominator

Factoring the denominator is an essential step in setting up partial fractions. It involves breaking down the polynomial into simpler polynomial factors.
Recognizing factorable forms, like quadratic or cubic expressions, allows for efficient breakdown:

• Accurately factored denominators set the stage for determining the structure of each term in the partial fraction.
• This understanding aids in decomposing the proper rational function into individual parts, matched with appropriate numerators.

In our exercise, the denominator is factored as \((x - 1)(x^2 + 1)\), enabling us to set up the next steps toward partial fraction decomposition.