To find the partial fraction decomposition of the given rational function:

$\frac{-2x^2 + 3x - 1}{x^4 - 2x^3 + 2x - 1},$

we need to follow these steps:

Step 1: Factorize the denominator

The denominator $x^4 - 2x^3 + 2x - 1$ should be factored into irreducible polynomials or linear terms.

Let us attempt to factorize:

• Using synthetic division or factoring techniques, this can often break into simpler components. I will compute this explicitly.

Step 2: Write the partial fraction form

Assume the factored denominator has terms like $(x - r)$, $(x - r)^k$, or quadratic terms. Express the fraction as: $\frac{-2x^2 + 3x - 1}{\text{(denominator factors)}} = \frac{A}{x - r_1} + \frac{B}{x - r_2} + \cdots$

Step 3: Solve for coefficients

Equate the expanded partial fractions to the original rational function and solve for $A$, $B$, etc., by matching coefficients or substituting specific values of $x$.

Let me compute the detailed solution for this specific function. sympy import symbols, factor, apart

Define the variables and the rational function

x = symbols('x') numerator = -2x**2 + 3x - 1 denominator = x4 - 2*x3 + 2*x - 1

Factorize the denominator

factored_denominator = factor(denominator)

Perform partial fraction decomposition

partial_fraction = apart(numerator / denominator)

factored_denominator, partial_fraction### Solution:

The denominator $x^4 - 2x^3 + 2x - 1$ factorizes as: $(x - 1)^3 (x + 1)$

The partial fraction decomposition of the given rational function is: $\frac{-2x^2 + 3x - 1}{x^4 - 2x^3 + 2x - 1} = \frac{3}{4(x + 1)} - \frac{3}{4(x - 1)} - \frac{1}{2(x - 1)^2}$

This expression represents the function in terms of its partial fractions.

Do you want detailed steps on how the coefficients were derived? Let me know if you have further questions!

Related Questions:

1. What is the role of factoring in partial fraction decomposition?
2. How does the order of the numerator and denominator affect decomposition?
3. Can this decomposition be verified by adding the fractions back together?
4. How would this change if the denominator included irreducible quadratic terms?
5. How is synthetic division used when factoring high-order polynomials?

Tip:

Always ensure that the degree of the numerator is less than the degree of the denominator before applying partial fraction decomposition. Use polynomial division if needed.