make into one fraction:
A+(B/(x-2))+(C/(x-2)^2)

To combine the expression \( A + \frac{B}{x-2} + \frac{C}{(x-2)^2} \) into a single fraction, follow these steps:

### Step 1: Find the common denominator
The common denominator for the terms is \( (x-2)^2 \).

### Step 2: Rewrite each term with the common denominator
- \( A = \frac{A(x-2)^2}{(x-2)^2} \)
- \( \frac{B}{x-2} = \frac{B(x-2)}{(x-2)^2} \)
- \( \frac{C}{(x-2)^2} \) is already in the form with denominator \( (x-2)^2 \)

### Step 3: Combine the numerators
Now, add the numerators over the common denominator:
\[
\frac{A(x-2)^2 + B(x-2) + C}{(x-2)^2}
\]

### Final Answer:
\[
\frac{A(x-2)^2 + B(x-2) + C}{(x-2)^2}
\]

This is the expression as a single fraction.

Would you like more details on simplifying or expanding this expression?

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Here are 5 related questions:
1. How would you simplify the numerator of this fraction?
2. What happens when \( x = 2 \) in this expression?
3. How can this type of rational function be applied in calculus?
4. How do you perform partial fraction decomposition on similar expressions?
5. What are the roots of the numerator when expanded?

**Tip**: When combining fractions, always make sure all terms have the same denominator before adding them together.

make into one fraction: A+(B/(x-2))+(C/(x-2)^2)

This guide explains how to combine the expression A + B/(x-2) + C/(x-2)^2 into a single fraction. Follow a step-by-step approach to find the common denominator, rewrite the terms, and combine them over a single denominator. Suitable for students in Grades 9-11, this problem focuses on rational expressions and algebraic manipulation.

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