Question

Write the partial fraction decomposition for the expression. $$ \frac{6 x^{2}-5 x}{(x+2)^{3}} $$

Short answer

The partial fraction decomposition of the given function is \(6x - 17 - 3/(x+2) - 13/(x+2)^2 + 22/(x+2)^3\).

Step-by-step solution

Polynomial Division

Since the degree of the numerator (\(6x^2 - 5x\)) is equal to the degree of the denominator (\((x+2)^3\)), first perform polynomial division. Dividing the numerator by the denominator, we get \(6x - 17\) as quotient and \(-34x + 34\) as remainder.

Partial fraction decomposition of the remainder

Express the remainder as the sum of simpler fractions. This gives \(-34x + 34 = A/(x+2) + B/(x+2)^2 + C/(x+2)^3\). Then, clear the fraction by multiplying through by \((x+2)^3\), which gives \(-34x + 34 = A(x+2)^2 + B(x+2) + C\). Now, identify the coefficients for A, B, and C.

Solving for A, B, and C

The coefficients for A, B, and C are found by setting x to values that simplify the equation. By setting x = -2, we can solve for C. To determine B, differentiate the equation, and again set x = -2. A is determined by equating the coefficients for the highest power of x in the equation, which results in A = -3, B = -13, and C = 22.

Writing the partial fraction decomposition

Finally, substitute A, B, and C into the partial fraction form yielding the decomposition -3/(x+2) - 13/(x+2)^2 + 22/(x+2)^3. Combine this with the quotient from Step 1 to get the entire decomposition as 6x - 17 - 3/(x+2) - 13/(x+2)^2 + 22/(x+2)^3.