Let's solve this problem step-by-step by applying the Remainder Theorem:
1. Find the remainder of [tex]\( f(x) \)[/tex] and [tex]\( x+5 \)[/tex].
According to the Remainder Theorem, if [tex]\( (x+c) \)[/tex] is a factor of [tex]\( f(x) \)[/tex], then [tex]\( f(-c) = 0 \)[/tex]. Given [tex]\( x+5 \)[/tex], we have [tex]\( c = -5 \)[/tex]. Evaluating [tex]\( f(-5) \)[/tex]:
Using [tex]\( f(x) = x^3 + 3x^2 - 25x - 75 \)[/tex],
[tex]\[
f(-5) = (-5)^3 + 3(-5)^2 - 25(-5) - 75 = -125 + 75 + 125 - 75 = 0
\][/tex]
So, the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x+5 \)[/tex] is [tex]\( 0 \)[/tex].
2. The result of this operation is [tex]\( 0 \)[/tex].
Since [tex]\( f(-5) = 0 \)[/tex], the result of this operation is [tex]\( 0 \)[/tex].
3. Therefore, [tex]\( (x+5) \)[/tex] is a factor of function [tex]\( f \)[/tex].
Since the remainder is [tex]\( 0 \)[/tex], [tex]\( (x+5) \)[/tex] is indeed a factor of [tex]\( f(x) \)[/tex].
4. So [tex]\( f( \; -5 \; ) = 0 \)[/tex].
We verified that [tex]\( f(-5) = 0 \)[/tex].
The correct answers for the drop-down menus are:
1. Find the remainder of [tex]\( f(x) \)[/tex] and [tex]\( x+5 \)[/tex].
2. The result of this operation is [tex]\( 0 \)[/tex].
3. Therefore, [tex]\( (x+5) \)[/tex] is a factor of function [tex]\( f \)[/tex].
4. So [tex]\( f( \; -5 \; ) = 0 \)[/tex].
