To solve the expression $\frac{a}{b} - \frac{b}{a}$, let's break it down step-by-step.

Step 1: Combine terms

We want to combine the two fractions, $\frac{a}{b}$ and $\frac{b}{a}$, into a single expression. To do this, we first find a common denominator.

The least common denominator (LCD) of $b$ and $a$ is $ab$, so we rewrite both fractions:

$\frac{a}{b} = \frac{a^2}{ab}, \quad \frac{b}{a} = \frac{b^2}{ab}$

Now subtract the two fractions:

$\frac{a}{b} - \frac{b}{a} = \frac{a^2}{ab} - \frac{b^2}{ab} = \frac{a^2 - b^2}{ab}$

Step 2: Factor the numerator

Notice that $a^2 - b^2$ is a difference of squares, which can be factored as:

$a^2 - b^2 = (a - b)(a + b)$

So the expression becomes:

$\frac{a}{b} - \frac{b}{a} = \frac{(a - b)(a + b)}{ab}$

Thus, the expression $\frac{a}{b} - \frac{b}{a}$ can indeed be written as the product of two factors: $(a - b)$ and $(a + b)$, and the denominator is $ab$.

Step 3: Consider the sum of the factors

The sum of the two factors is:

$\frac{a}{b} + \frac{b}{a}$

We follow a similar process to combine the two fractions:

$\frac{a}{b} + \frac{b}{a} = \frac{a^2}{ab} + \frac{b^2}{ab} = \frac{a^2 + b^2}{ab}$

So the sum of the two factors is $\frac{a^2 + b^2}{ab}$.

Conclusion:

• $\frac{a}{b} - \frac{b}{a}$ can be written as the product of two factors: $\frac{(a - b)(a + b)}{ab}$.
• The sum of the two factors $\frac{a}{b} + \frac{b}{a}$ is $\frac{a^2 + b^2}{ab}$.

Let me know if you need more clarification or have further questions!

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Here are 5 related questions:

1. What is the difference of squares formula, and how does it simplify expressions?
2. How do you simplify expressions involving fractions with different denominators?
3. How can you factor other quadratic expressions, such as $x^2 + 6x + 9$?
4. What is the relationship between addition and subtraction of fractions with the same denominator?
5. How would the expressions change if $a$ and $b$ were negative numbers?

Tip: Always look for patterns like the difference of squares or perfect square trinomials to simplify algebraic expressions!