To divide \( x^2 - a^2 \) by \( x - a \), we can first recognize that \( x^2 - a^2 \) can be factored using the difference of squares formula:
\[ x^2 - a^2 = (x - a)(x + a) \]
Now, when we divide \( x^2 - a^2 \) by \( x - a \), we can write:
\[ \frac{x^2 - a^2}{x - a} = \frac{(x - a)(x + a)}{x - a} \]
As long as \( x \neq a \), we can cancel the \( x - a \) term:
\[ \frac{(x - a)(x + a)}{x - a} = x + a \]
Thus, we can say that:
\[ \frac{x^2 - a^2}{x - a} = x + a \quad \text{for } x \neq a \]
This gives us polynomial identities that express the relationship between these expressions. Specifically, we have the following identities:
- Factorization Identity: \( x^2 - a^2 = (x - a)(x + a) \)
- Division Identity: For \( x \neq a \), \( \frac{x^2 - a^2}{x - a} = x + a \)
If you use this result in any application involving \( x^2 - a^2 \) and \( x - a \), it emphasizes the relationship between polynomial division and factoring.