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]
