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To solve this, let's analyze the expression given and determine the number of first-type partial fractions in its decomposition.

### Given Expression:
\[
\frac{6x}{x^3 + 8}
\]

We are asked about the number of "first-type" partial fractions. Typically, first-type partial fractions refer to those with simple linear denominators.

### Step-by-Step Solution:
1. **Factor the Denominator**: We need to factor \(x^3 + 8\) first.
   - Recognize \(x^3 + 8\) as a sum of cubes: \(x^3 + 8 = x^3 + 2^3\).
   - Using the sum of cubes formula, \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\), we get:
     \[
     x^3 + 8 = (x + 2)(x^2 - 2x + 4)
     \]

2. **Set Up Partial Fraction Decomposition**:
   Since \(x + 2\) is a linear factor and \(x^2 - 2x + 4\) is an irreducible quadratic factor, the partial fraction decomposition will take the form:
   \[
   \frac{6x}{(x + 2)(x^2 - 2x + 4)} = \frac{A}{x + 2} + \frac{Bx + C}{x^2 - 2x + 4}
   \]
   Here, \( \frac{A}{x + 2} \) represents the partial fraction of the "first type" since it has a simple linear denominator.

3. **Count the First-Type Fractions**:
   In this decomposition, there is **one partial fraction of the first type**: \( \frac{A}{x + 2} \).

### Answer:
The answer is:
\[
\text{1}
\]

Would you like a further breakdown of the solution or have any other questions?

### Related Questions:
1. What are the steps to decompose a fraction with irreducible quadratic terms in the denominator?
2. How can we determine if a quadratic factor in the denominator is irreducible?
3. What defines a "first-type" partial fraction in decomposition?
4. How does the decomposition change if the denominator has repeated linear factors?
5. How do we solve for constants \(A\), \(B\), and \(C\) in the partial fraction decomposition?

### Tip:
Remember that identifying the type of factors in the denominator is key to setting up the correct form of partial fractions.

Při rozkladu výrazu \(\frac{6x}{x^3 + 8}\) na parciální zlomky, kolik bude ve formálním tvaru rozkladu tohoto výrazu parciálních zlomků prvního typu? (Odpověď zadejte jako celé číslo.)

Learn how to decompose the expression \(\frac{6x}{x^3 + 8}\) into partial fractions and determine the number of first-type partial fractions. This solution covers key algebraic techniques, including factoring cubic expressions and partial fraction decomposition, suitable for high school students.

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