Certainly, let's determine if [tex]\( x + 10 \)[/tex] is a factor of the function [tex]\( f(x) = x^3 - 75x + 250 \)[/tex] using the Remainder Theorem.
### Step-by-Step Solution:
1. Understanding the Remainder Theorem:
The Remainder Theorem states that for a polynomial [tex]\( f(x) \)[/tex], if we divide [tex]\( f(x) \)[/tex] by [tex]\( x - c \)[/tex], the remainder of this division is [tex]\( f(c) \)[/tex]. As a direct application, [tex]\( x - c \)[/tex] is a factor of [tex]\( f(x) \)[/tex] if and only if [tex]\( f(c) = 0 \)[/tex].
2. Identify the values in the problem:
We need to check if [tex]\( x + 10 \)[/tex] is a factor of [tex]\( f(x) = x^3 - 75x + 250 \)[/tex].
- Here, [tex]\( x + 10 \)[/tex] can be rewritten as [tex]\( x - (-10) \)[/tex].
- According to the Remainder Theorem, to check if [tex]\( x + 10 \)[/tex] is a factor, we need to evaluate [tex]\( f(-10) \)[/tex].
3. Substitute [tex]\( x = -10 \)[/tex] into the polynomial:
[tex]\[
f(-10) = (-10)^3 - 75(-10) + 250
\][/tex]
4. Perform the calculations step-by-step:
- Calculate [tex]\((-10)^3\)[/tex]:
[tex]\[
(-10)^3 = -1000
\][/tex]
- Calculate [tex]\(-75 \times (-10)\)[/tex]:
[tex]\[
-75 \times (-10) = 750
\][/tex]
- Now, sum these values including the constant 250:
[tex]\[
f(-10) = -1000 + 750 + 250
\][/tex]
5. Combine the results:
[tex]\[
-1000 + 750 + 250 = 0
\][/tex]
6. Conclusion:
Since [tex]\( f(-10) = 0 \)[/tex], by the Remainder Theorem, it indicates that [tex]\( x + 10 \)[/tex] is indeed a factor of the polynomial [tex]\( f(x) = x^3 - 75x + 250 \)[/tex].
Thus, we have shown that [tex]\( x + 10 \)[/tex] is a factor of the function [tex]\( f(x) = x^3 - 75x + 250 \)[/tex] because the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( x + 10 \)[/tex] is zero.
