To solve the given problem step-by-step:
1. Define the Polynomials:
Given [tex]\( f(x) = x^2 - 8x + 15 \)[/tex] and [tex]\( g(x) = x - 3 \)[/tex].
2. Perform the Division:
We need to divide [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex].
[tex]\[
\frac{f(x)}{g(x)} = \frac{x^2 - 8x + 15}{x - 3}
\][/tex]
3. Simplification:
Factorize [tex]\( f(x) \)[/tex] if possible.
[tex]\[
f(x) = x^2 - 8x + 15 = (x - 3)(x - 5)
\][/tex]
So,
[tex]\[
\frac{(x - 3)(x - 5)}{x - 3}
\][/tex]
Here, [tex]\( x \neq 3 \)[/tex] to avoid division by zero. Thus,
[tex]\[
h(x) = x - 5 \quad \text{for} \quad x \neq 3
\][/tex]
4. Find the Domain:
[tex]\( h(x) \)[/tex] is [tex]\( x - 5 \)[/tex] but it is undefined at [tex]\( x = 3 \)[/tex] (because [tex]\( g(x) = 0 \)[/tex] at [tex]\( x = 3 \)[/tex]). Therefore, the domain of [tex]\( h(x) \)[/tex] excludes [tex]\( x = 3 \)[/tex].
The final answers are:
- [tex]\( h(x) = x - 5 \)[/tex]
- The domain of [tex]\( h(x) \)[/tex] is [tex]\((-\infty, 3) \cup (3, \infty)\)[/tex]
Thus, the correctly completed statements are:
- [tex]\( h(x)= \textbf{x - 5} \)[/tex]
- The domain of [tex]\( h(x) \)[/tex] is [tex]\(\textbf{(-\infty, 3)}\)[/tex] U [tex]\(\textbf{(3, \infty)}\)[/tex]
