Answered

Given the functions [tex]f(x)=x^3+6x^2+x^{\frac{1}{2}}[/tex] and [tex]g(x)=x^{\frac{1}{2}}[/tex], find [tex]f(x) \div g(x)[/tex].

A. [tex]x^{\frac{5}{2}}+6x^{\frac{3}{2}}-1[/tex]

B. [tex]\left(-x^5+6x^3+1\right)^{\frac{1}{2}}[/tex]

C. [tex]x^{\frac{5}{2}}+6x^{\frac{3}{2}}+1[/tex]

D. [tex]x^{\frac{3}{2}}+6x+x^{\frac{1}{4}}[/tex]

E. [tex]x^{\frac{3}{2}}-6x+x^{\frac{1}{4}}[/tex]



Answer :

GinnyAnswer
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]

Other Questions