[tex]f(x) \div g(x)=3 x^{2}+x+2+\frac{3}{5 x-1}[/tex]
Solution:
Given:
[tex]f(x)=15 x^{3}+2 x^{2}+9 x+1[/tex]
[tex]g(x)=5 x-1[/tex]
[tex]f(x) \div g(x)[/tex]
[tex]$=\frac{\left(15 x^{3}+2 x^{2}+9 x+1\right)}{5 x-1}[/tex]
If we divide [tex]\frac{15 x^{3}}{5 x}=3 x^{2}[/tex] quotient and remainder is [tex]5 x^{2}+9 x+1[/tex].
[tex]$=3 x^{2}+\frac{5 x^{2}+9 x+1}{5 x-1}[/tex]
If we divide [tex]\frac{5 x^{2}}{5 x}=x[/tex] quotient and remainder is [tex]10 x+1[/tex].
[tex]$=3 x^{2}+x+\frac{10 x+1}{5 x-1}[/tex]
If we divide [tex]\frac{10 x}{5 x}=2[/tex] quotient and remainder is 3.
[tex]$=3 x^{2}+x+2+\frac{3}{5 x-1}[/tex].
Therefore, [tex]f(x) \div g(x)=3 x^{2}+x+2+\frac{3}{5 x-1}[/tex].
