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Divide [tex]$f(x)$[/tex] by [tex]$d(x)$[/tex]. Your answer should be in the following format:

[tex]\[
\begin{array}{c}
\frac{f(x)}{d(x)} = Q(x) + \frac{R(x)}{d(x)} \\
\frac{f(x)}{d(x)} = \frac{x^3 + 5x^2 + 5x - 1}{x + 3} \\
R(x) = [?]
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
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].
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