Write out the form of the partial fraction decomposition of the function (see example). Do not determine the numerical values of the coefficients.
(a)    
x4
(x3 + x)(x2 − x + 8)



To write the partial fraction decomposition of the given function:

\[
\frac{x^4}{(x^3 + x)(x^2 - x + 8)}
\]

### Step-by-step process:

1. **Factor the denominator:**
   - The denominator can be factored into two parts:
     - \(x(x^2 + 1)\), which comes from \(x^3 + x\)
     - \(x^2 - x + 8\)

   Therefore, the function is now:
   \[
   \frac{x^4}{x(x^2 + 1)(x^2 - x + 8)}
   \]

2. **Set up the partial fraction decomposition:**
   - For the linear term \(x\), the partial fraction will take the form: \(\frac{A}{x}\).
   - For the quadratic term \(x^2 + 1\), since it is irreducible, the partial fraction will take the form: \(\frac{Bx + C}{x^2 + 1}\).
   - For the quadratic term \(x^2 - x + 8\), since it is also irreducible, the partial fraction will take the form: \(\frac{Dx + E}{x^2 - x + 8}\).

3. **Write the decomposition:**
   The partial fraction decomposition of the given function is:

   \[
   \frac{x^4}{x(x^2 + 1)(x^2 - x + 8)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1} + \frac{Dx + E}{x^2 - x + 8}
   \]

This is the form of the partial fraction decomposition without determining the numerical values of the coefficients \(A\), \(B\), \(C\), \(D\), and \(E\).

---

Would you like further details on how the coefficients are determined?

### Related questions:
1. How do you solve for the coefficients in partial fraction decompositions?
2. What happens when a quadratic factor in the denominator can be factored further?
3. How do you decompose a fraction with a repeated linear factor in the denominator?
4. What are the conditions for a factor to be considered irreducible?
5. How would the decomposition change if the numerator had a higher degree than the denominator?

**Tip:** Always check if the degree of the numerator is less than the degree of the denominator before starting the decomposition. If not, perform polynomial division first.

Write out the form of the partial fraction decomposition of the function x^4 / ((x^3 + x)(x^2 - x + 8)). Do not determine the numerical values of the coefficients.

Learn how to write the partial fraction decomposition of the function x^4 / ((x^3 + x)(x^2 - x + 8)) without determining the coefficients. This detailed guide covers concepts such as polynomial factorization, irreducible quadratics, and the structure of partial fraction decompositions. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation