Write out the general form of the partial fraction decomposition of the function. Do not determine the numerical values of the constants.

Question

Answer 1

Since the degree of the numerator is not less than the degree of the denominator, we first divide to get:

x² + (5x - 4) / (x - 1)² = x² + A / (x-1) + B / (x-1)²

Answer 2

Let us take a closer look at the denominator x² - 2x + 1

Using the quadratic identity

(a - b)² = a² - 2ab + b²

we can write this denominator as

x² - 2x + 1 = (x - 1)²

The highest term in the numerator is x⁴, of order 4.

The highest term in the denominator is x², of order 2.

The highest term in the quotient will thus also be of order 2 (because x⁴ / x² = x²)

Ax² where A is a real number (the A here is actually 1 but since the instructions were to write the general form, we can keep it as A)

The next highest term in the quotient would be of order 1 (1 less than the previous term's order).

Bx where B is a real number

Note: it might turn out that B = 0 and that there is no term of order 1 in the quotient, which would be fine; we still need to account for it.

The next highest term in the quotient would be of order 0 (1 less than the previous term's order), so just a constant.

C where C is a real number

Note: it might turn out that C = 0 and that there is no term of order 0 in the quotient, which would be fine; we still need to account for it.

Finally, since the denominator can be written as (x - 1)², we need two fractions for the remainder:

D / (x - 1) and

E / (x - 1)²

where D and E are real numbers

The general form of the partial fraction decomposition of the function

(x⁴ - 2x³ + x² + 5x - 4) / (x² - 2x + 1)

is thus given by

Ax² + Bx + C + D / (x - 1) + E / (x - 1)²

where A, B, C, D, and E are real numbers.

Edit: due to the Homework system deeming this answer incorrect, here is an updated answer:

(x⁴ - 2x³ + x² + 5x - 4) / (x² - 2x + 1)

= (x⁴ - 2x³ + x²) / (x² - 2x + 1) + (5x - 4) / (x² - 2x + 1)

= x²(x² - 2x + 1) / (x² - 2x + 1) + (5x - 4) / (x² - 2x + 1)

= x² + (5x - 4) / (x² - 2x + 1)

Since x² - 2x + 1 = (x - 1)²

the last fraction can be decomposed into

A / (x - 1) + B / (x - 1)²

so that the general decomposition is

x² + A / (x - 1) + B / (x - 1)²

where A and B are real numbers