Write out the form of the partial fraction decomposition of the function (as in this example). do not determine the numerical values of the coefficients. (a) x4 − 2x3 + x2 + 3x − 2 x2 − 2x + 1

The form of the partial fraction decomposition of the function (x⁴ − 2x³ + x² + 3x − 2) ÷ x² − 2x + 1 without determining the numerical values of the coefficient is; x² + a/(x - 1) + b/(x - 1)²

The polynomial division of the function;

(x⁴ - 2x³ + x² + 3x -2)/(x² - 2x + 1). is as shown in the attached image.

Therefore, the result of the division is;

x² + (3x - 2)/(x² - 2x + 1)

However, because the function (x² - 2x + 1) can be expressed as a square in the form;

(x - 1)², so that we have;

x² + (3x - 2)/(x - 1)²

The general rule for partial fractions with repeated roots as in this case allows that the express be written as;

x² + a/(x - 1) + b/(x - 1)²

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The degree of the numerator exceeds the degree of the denominator, so first you have to divide:

Now, , so the remainder term can be expanded to get

d

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