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To solve this, we are asked to prove the identity:

\[
\frac{1}{(x - a)(x - b)} = \frac{1}{a - b} \left( \frac{1}{x - a} - \frac{1}{x - b} \right)
\]
where \( x \neq a \), \( x \neq b \), and \( a \neq b \).

### Proof:

1. Start with the right-hand side:
   \[
   \frac{1}{a - b} \left( \frac{1}{x - a} - \frac{1}{x - b} \right)
   \]
   We need to simplify the expression inside the parentheses first. 

2. Find a common denominator for the terms inside the parentheses:
   \[
   \frac{1}{x - a} - \frac{1}{x - b} = \frac{(x - b) - (x - a)}{(x - a)(x - b)}
   \]
   Simplify the numerator:
   \[
   (x - b) - (x - a) = x - b - x + a = a - b
   \]
   Therefore:
   \[
   \frac{1}{x - a} - \frac{1}{x - b} = \frac{a - b}{(x - a)(x - b)}
   \]

3. Now substitute this result back into the original expression:
   \[
   \frac{1}{a - b} \cdot \frac{a - b}{(x - a)(x - b)}
   \]
   The \( a - b \) terms cancel out:
   \[
   \frac{1}{(x - a)(x - b)}
   \]

Thus, we have shown that the left-hand side is equal to the right-hand side:

\[
\frac{1}{(x - a)(x - b)} = \frac{1}{a - b} \left( \frac{1}{x - a} - \frac{1}{x - b} \right)
\]

This completes the proof.

Do you need further details on any step, or have any questions? Here are five related questions to expand your understanding:

1. How would this proof change if we had three distinct factors in the denominator?
2. Can you apply a similar approach to simplify rational expressions with different forms of quadratic denominators?
3. How do partial fractions relate to this type of identity?
4. What happens when \( a = b \), and why is this condition excluded?
5. How does this proof apply to integrals involving similar rational functions?

**Tip:** When working with complex rational expressions, always look for common factors or terms that can simplify, especially when you're working with fractions involving multiple variables.

Show that \( \frac{1}{(x - a)(x - b)} = \frac{1}{a - b} \left( \frac{1}{x - a} - \frac{1}{x - b} \right) \) if \(x \neq a\), \(x \neq b\), and \(a \neq b\).

Learn how to prove the identity \( \frac{1}{(x - a)(x - b)} = \frac{1}{a - b} \left( \frac{1}{x - a} - \frac{1}{x - b} \right) \), a key concept in simplifying rational expressions using partial fractions. This proof is ideal for students in Grades 10-12, covering advanced algebra and rational expressions.

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