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Resolve into partial fraction: (8x^2 - 19x - 24) / (x - 1)(x^2 - 2x - 5)

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Final answer:

To resolve the given function into partial fractions, we assume it can be broken down into a specific form, then multiply through by the denominator and equate coefficients to solve for the variables that will represent each part of the sum.

Step-by-step explanation:

The student asked how to resolve the function (8x^2 - 19x - 24) / (x - 1)(x^2 - 2x - 5) into partial fractions. To do this, we first assume that the function can be written as a sum of fractions of the form:

A/(x - 1) + (Bx + C)/(x^2 - 2x - 5)

Next, we multiply both sides by the denominator (x - 1)(x^2 - 2x - 5) to get rid of the fractions:

8x^2 - 19x - 24 = A(x^2 - 2x - 5) + (Bx + C)(x - 1)

We then expand the right side and collect like terms to match the coefficients on both sides of the equation. By equating the coefficients of x^2, x, and the constant term, we obtain a system of equations that will allow us to solve for A, B, and C. Lastly, we use these values to write the original function as a sum of its partial fractions.

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