Question

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

Answer 1

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

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

$x^2+\dfrac{3x-2}{x^2-2x+1}$

Now, $x^2-2x+1=(x-1)^2$, so the remainder term can be expanded to get

$\boxed{x^2+\dfrac a{x-1}+\dfrac b{(x-1)^2}}$

Answer 2

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

• From observation, the numerator of the function is a quartic function while the denominator is a quadratic function; there's a need to first perform the polynomial division.

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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