Answer:
[tex]\frac{f(x)}{g(x)}=x^{\frac{5}{2}} +6 x^{\frac{3}{2}} +1}[/tex]
Step-by-step explanation:
[tex]f(x)=x^3+6x^2+\sqrt{x}[/tex]
[tex]g(x)= \sqrt{x}[/tex]
we are asked to determine
[tex]\frac{f(x)}{g(x)} [/tex]
Let us do it step by step.
[tex]f(x)=x^3+6x^2+\sqrt{x}[/tex]
[tex]f(x)=x^2 \times x+6x \times x +\sqrt{x}[/tex]
[tex]f(x)=x^2 \times \sqrt{x} \times \sqrt{x} +6x \times \sqrt{x} \times \sqrt{x} +\sqrt{x}[/tex]
Taking [tex]\sqrt{x}[/tex] as GCF
[tex]f(x)= \sqrt{x}(x^2 \times \sqrt{x} +6x \times \sqrt{x} +1) [/tex]
Hence
[tex]\frac{f(x)}{g(x)}=\frac{\sqrt{x}(x^2 \times \sqrt{x} +6x \times \sqrt{x} +1)}{ \sqrt{x}}[/tex]
[tex]\frac{f(x)}{g(x)}=x^2 \times \sqrt{x} +6x \times \sqrt{x} +1[/tex]
[tex]\sqrt{x}=x^\frac{1}{2}[/tex]
[tex]\frac{f(x)}{g(x)}=x^2 \times x^{\frac{1}{2}} +6 \times x \times x^{\frac{1}{2}} +1[/tex]
Using law of exponents
[tex]a^m \times a^n = a^{m+n}[/tex]
[tex]\frac{f(x)}{g(x)}=x^{2+\frac{1}{2}} +6 \times x^{1+\frac{1}{2}} +1}[/tex]
[tex]\frac{f(x)}{g(x)}=x^{\frac{5}{2}} +6 x^{\frac{3}{2}} +1}[/tex]
