Question

Write the appropriate form for finding the partial fractions of

x¹¹− 3⁴x+ 9x + 2 / x^3(x² − 4)(x² + 5x + 10)²

Answer 1

To find the partial fractions, we need to decompose the expression into simpler fractions and solve for unknown constants.

Step-by-step explanation:

To find the partial fractions of the given expression, we need to decompose it into simpler fractions. First, we factor the denominator as x^3(x^2 - 4)(x^2 + 5x + 10)^2. The first step is to consider the factored denominators and set up the partial fractions with unknown constants A, B, C, D, E, and F:

• x^3: A/x
• (x^2 - 4): B/(x^2 - 4)
• (x^2 + 5x + 10)^2: (Cx + D)/(x^2 + 5x + 10) + (Ex + F)/(x^2 + 5x + 10)^2

Next, we multiply both sides of the equation by the common denominator to eliminate the fractions and solve for the unknown constants. Once we have the values for A, B, C, D, E, and F, we can substitute them back into the partial fractions to get the complete decomposition.