Certainly! Let's walk through the steps to verify the statement using the Remainder Theorem.
1. Find the remainder of [tex]\(f(x)\)[/tex] divided by [tex]\(x+5\)[/tex].
To check if [tex]\(x+5\)[/tex] is a factor of [tex]\(f(x) = x^3 + 3x^2 - 25x - 75\)[/tex], we will use the Remainder Theorem. According to the Remainder Theorem, the remainder of dividing [tex]\(f(x)\)[/tex] by [tex]\(x + 5\)[/tex] is [tex]\(f(-5)\)[/tex].
2. The value of [tex]\(f(-5)\)[/tex]:
[tex]\[
f(-5) = (-5)^3 + 3(-5)^2 - 25(-5) - 75
\][/tex]
[tex]\[
= -125 + 3(25) + 125 - 75
\][/tex]
[tex]\[
= -125 + 75 + 125 - 75
\][/tex]
[tex]\[
= 0
\][/tex]
The result of this operation is 0.
3. Therefore, [tex]\((x + 5)\)[/tex] is a factor of function [tex]\(f(x)\)[/tex].
Since [tex]\(f(-5) = 0\)[/tex], by the Remainder Theorem, [tex]\((x + 5)\)[/tex] is indeed a factor of [tex]\(f(x)\)[/tex].
4. So, [tex]\(f(-5) = 0\)[/tex].
Putting these steps in the blanks:
1. Find the remainder of [tex]\(f(x)\)[/tex] and [tex]\(x+5\)[/tex].
2. The result of this operation is 0.
3. Therefore, [tex]\((x+5)\)[/tex] is a factor of function [tex]\(f(x)\)[/tex].
4. So [tex]\(f(-5) = 0\)[/tex].
This verifies that [tex]\((x+5)\)[/tex] is a factor of [tex]\(f(x)\)[/tex].
