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

[tex]\[
\begin{array}{c}
\frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\
\frac{f(x)}{d(x)}=\frac{x^4-35 x^2}{x+6} \\
R(x)=[?]
\end{array}
\][/tex]



Answer :

To divide the polynomial [tex]\( f(x) \)[/tex] by [tex]\( d(x) \)[/tex], we aim to express the division in the following format:
[tex]\[ \frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \][/tex]
where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R(x) \)[/tex] is the remainder.

Given the polynomials:
[tex]\[ f(x) = x^4 - 35x^2 \][/tex]
[tex]\[ d(x) = x + 6 \][/tex]

When we perform the polynomial division of [tex]\( f(x) \)[/tex] by [tex]\( d(x) \)[/tex], we obtain:
[tex]\[ Q(x) = x^3 - 6x^2 + x - 6 \][/tex]
[tex]\[ R(x) = 36 \][/tex]

Thus, our division can be represented as:
[tex]\[ \frac{x^4 - 35x^2}{x + 6} = \left( x^3 - 6x^2 + x - 6 \right) + \frac{36}{x + 6} \][/tex]

So, our final answer in the specified format is:
[tex]\[ \begin{array}{c} \frac{f(x)}{d(x)}=Q(x)+\frac{R(x)}{d(x)} \\ \frac{f(x)}{d(x)}=\frac{x^4-35 x^2}{x+6} \\ R(x)=[36] \end{array} \][/tex]