To solve [tex]\( f(x) \div g(x) \)[/tex], where [tex]\( f(x) = x^3 + 6x^2 + x^{\frac{1}{2}} \)[/tex] and [tex]\( g(x) = x^{\frac{1}{2}} \)[/tex]:
1. Write the division of the functions explicitly:
[tex]\[
\frac{f(x)}{g(x)} = \frac{x^3 + 6x^2 + x^{\frac{1}{2}}}{x^{\frac{1}{2}}}
\][/tex]
2. Divide each term in the numerator by the term in the denominator:
[tex]\[
\frac{f(x)}{g(x)} = \frac{x^3}{x^{\frac{1}{2}}} + \frac{6x^2}{x^{\frac{1}{2}}} + \frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}}
\][/tex]
3. Simplify each term individually:
- For the first term:
[tex]\[
\frac{x^3}{x^{\frac{1}{2}}} = x^{3 - \frac{1}{2}} = x^{\frac{6}{2} - \frac{1}{2}} = x^{\frac{5}{2}}
\][/tex]
- For the second term:
[tex]\[
\frac{6x^2}{x^{\frac{1}{2}}} = 6x^{2 - \frac{1}{2}} = 6x^{\frac{4}{2} - \frac{1}{2}} = 6x^{\frac{3}{2}}
\][/tex]
- For the third term:
[tex]\[
\frac{x^{\frac{1}{2}}}{x^{\frac{1}{2}}} = x^{\frac{1}{2} - \frac{1}{2}} = x^0 = 1
\][/tex]
4. Combine the simplified terms:
[tex]\[
\frac{f(x)}{g(x)} = x^{\frac{5}{2}} + 6x^{\frac{3}{2}} + 1
\][/tex]
Thus, the simplified result of [tex]\( \frac{f(x)}{g(x)} \)[/tex] is:
[tex]\[
\boxed{x^{\frac{5}{2}} + 6x^{\frac{3}{2}} + 1}
\][/tex]
Therefore, the correct answer is:
C. [tex]\( x^{\frac{5}{2}} + 6x^{\frac{3}{2}} + 1 \)[/tex]
