Final answer:
To express the rational expression (2x² + 6x - 6)/(x³ - 1) as a sum of partial fractions, we factor the denominator, set up the partial fraction decomposition, clear denominators to equate coefficients, and solve the resulting system of equations for the constants.
Step-by-step explanation:
To express a rational expression as a sum of partial fractions, we first identify the type of decomposition that the denominator suggests. For the given rational expression (2x² + 6x - 6)/(x³ - 1), we note that the denominator factors as (x - 1)(x² + x + 1), which is the difference of cubes. We can then express the original expression as a sum of partial fractions by finding constants A, B, and C such that:
(2x² + 6x - 6)/(x - 1)(x² + x + 1) = A/(x - 1) + (Bx + C)/(x² + x + 1)
To find A, B, and C, we must clear the denominators by multiplying both sides by the common denominator (x³ - 1), yielding an equation that allows us to equate coefficients and solve for the unknowns. Multiplying through and comparing the coefficients of like powers of x on both sides of the equation will give us a system of equations, which we must solve to determine the values of A, B, and C.
Once we have the values of A, B, and C, we can write the original rational expression as the sum of its partial fractions and integrate if necessary, which is typically the purpose of this decomposition in calculus.