91Ó°ÊÓ

When dealing with partial fraction decomposition, a repeated linear factor occurs when one of the factors in the denominator is raised to a power greater than one. In our exercise, the term \( z^2 \) represents a repeated linear factor. This means that the factor \( z \) is repeated, in this case, twice, hence \( z\cdot z = z^2 \).

Including a repeated linear factor in decomposition requires setting up multiple fractions; one for each power of the repeated factor:

• \( \frac{A}{z} \) for the first occurrence
• \( \frac{B}{z^2} \) for the second occurrence

This accounts for all instances of the factor. Each fraction has a constant that we need to determine through solving a system of equations. By understanding repeated linear factors, you can efficiently decompose and solve more complex rational expressions.

A non-repeated linear factor is a factor in the denominator that appears only once. In the given exercise, \((z-1)\) is a non-repeated factor.

When dealing with non-repeated linear factors in partial fraction decomposition, you set up a fraction with the factor in the denominator:

• \( \frac{C}{z-1} \)

This represents the unique linear component that does not repeat in the expression. Determining the constant \( C \) involves matching the numerators of the decomposed fractions to the original function's numerator and solving the resultant system of equations.

Properly recognizing non-repeated linear factors simplifies the process of obtaining the correct constants in decomposition, such as \( C \) in this example.

In partial fraction decomposition, after setting up several decomposed fractions, we often end up with a system of equations to solve. This system comes from equating the numerator of the original function with the combined numerators of the decomposed terms, as seen in the exercise.

Step-by-step, this involves:

• Expanding all terms in the numerators after multiplying through by the least common denominator.
• Collecting like terms across all powers of \( z \).

From our exercise:

• The terms \( Cz^2 + (A+B)z - (A+B) \) needed to equal \( z + 1 \)

This results in equations like:

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

By solving the system, you determine the constants needed for the decomposed expression. However, take care to verify assumptions and check for consistency to avoid errors as seen in step adjustments. Understanding this interrelation of equations is key to successful partial fraction decomposition.