To perform polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( d(x) \)[/tex], follow the format provided to present the solution.
1. Division Process:
Given polynomials:
[tex]\[ f(x) = x^3 + 5x^2 + 5x - 1 \][/tex]
[tex]\[ d(x) = x + 3 \][/tex]
When dividing these polynomials, we find the quotient [tex]\( Q(x) \)[/tex] and remainder [tex]\( R(x) \)[/tex].
2. Result:
[tex]\[ Q(x) = x^2 + 2x - 1 \][/tex]
[tex]\[ R(x) = 2 \][/tex]
Now we can express the division in the required format:
[tex]\[
\begin{array}{c}
\frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\
\frac{f(x)}{d(x)}=\frac{x^3 + 5x^2 + 5x - 1}{x + 3} \\
R(x)=[2]
\end{array}
\][/tex]
Hence, the remainder [tex]\( R(x) \)[/tex] is [tex]\( 2 \)[/tex].
