To use the remainder theorem to find [tex]\( F(-1) \)[/tex] for the polynomial function [tex]\( F(x) = -x^3 + 6x^2 - 4x + 11 \)[/tex]:
1. Substitute [tex]\( x = -1 \)[/tex] into the polynomial:
Given the function [tex]\( F(x) = -x^3 + 6x^2 - 4x + 11 \)[/tex], we need to evaluate [tex]\( F(-1) \)[/tex].
2. Evaluate each term:
- For the term [tex]\( -x^3 \)[/tex]:
[tex]\[
-(-1)^3 = -(-1) = 1
\][/tex]
- For the term [tex]\( 6x^2 \)[/tex]:
[tex]\[
6(-1)^2 = 6(1) = 6
\][/tex]
- For the term [tex]\( -4x \)[/tex]:
[tex]\[
-4(-1) = 4
\][/tex]
- For the constant term [tex]\( +11 \)[/tex]:
[tex]\[
11
\][/tex]
3. Add the results of each term:
[tex]\[
F(-1) = 1 + 6 + 4 + 11
\][/tex]
4. Combine the results:
[tex]\[
F(-1) = 22
\][/tex]
Hence, the result of using the remainder theorem to find [tex]\( F(-1) \)[/tex] for the polynomial function [tex]\( F(x) = -x^3 + 6x^2 - 4x + 11 \)[/tex] is [tex]\(\boxed{22}\)[/tex].
