Question

what polynomial identities would result from dividing x^2-a^2 by x-a

Answer 1

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:

1. **Factorization Identity**: \( x^2 - a^2 = (x - a)(x + a) \)
2. **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.