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To use the **cover-up rule** in partial fraction decomposition, it's best suited for cases where the denominator has distinct linear factors. Here's a solution guide for the questions where this method is applicable from the image you provided:

### Let's solve Question 1 as an example.

1. **Separate \( \frac{x}{x^2 + 5x + 6} \) into partial fractions.**

   First, factor the denominator:
   \[
   x^2 + 5x + 6 = (x + 2)(x + 3)
   \]

   So we set up the equation:
   \[
   \frac{x}{(x + 2)(x + 3)} = \frac{A}{x + 2} + \frac{B}{x + 3}
   \]

   Multiply both sides by \( (x + 2)(x + 3) \):
   \[
   x = A(x + 3) + B(x + 2)
   \]

   To solve for \( A \) and \( B \) using the cover-up method:
   
   - For \( A \): cover up \( x + 2 \) in the original expression \( \frac{x}{(x + 2)(x + 3)} \) and substitute \( x = -2 \):
     \[
     A = \frac{-2}{-2 + 3} = \frac{-2}{1} = -2
     \]
   - For \( B \): cover up \( x + 3 \) in the original expression and substitute \( x = -3 \):
     \[
     B = \frac{-3}{-3 + 2} = \frac{-3}{-1} = 3
     \]

   Thus, the partial fraction decomposition is:
   \[
   \frac{x}{(x + 2)(x + 3)} = \frac{-2}{x + 2} + \frac{3}{x + 3}
   \]

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Would you like to see the solutions for additional questions from the list?

### Related Questions:
1. How can we apply the cover-up rule to a partial fraction with repeated linear factors, as in Question 3?
2. What is the decomposition of a fraction with a quadratic factor in the denominator, as seen in Question 4?
3. How can we approach partial fractions when the degree of the numerator is higher than the denominator, like in Question 9?
4. How does the cover-up method differ from long division in partial fraction decomposition?
5. How can the cover-up rule be applied to partial fractions with complex roots?

### Tip:
The cover-up rule is only effective when there are non-repeated linear factors in the denominator. For repeated or quadratic factors, other techniques are needed.

Explore the method of partial fraction decomposition with the cover-up rule. This guide solves an example problem, decomposing a rational function by covering up factors to find coefficients. Suitable for Grades 10-12 students learning algebra and partial fractions.

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