Let's tackle the first part of the question. Given the function [tex]\( f(x) = x - 9 \)[/tex], we need to find [tex]\( f(x) - f(-x) \)[/tex].
1. Calculate [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = x - 9 \][/tex]
2. Calculate [tex]\( f(-x) \)[/tex]:
Replace [tex]\( x \)[/tex] with [tex]\(-x\)[/tex] in the function [tex]\( f \)[/tex]:
[tex]\[ f(-x) = -x - 9 \][/tex]
3. Subtract [tex]\( f(-x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) - f(-x) = (x - 9) - (-x - 9) \][/tex]
[tex]\[ f(x) - f(-x) = (x - 9) + x + 9 \][/tex]
[tex]\[ f(x) - f(-x) = x + x - 9 + 9 \][/tex]
[tex]\[ f(x) - f(-x) = 2x \][/tex]
So, the result is [tex]\( f(x) - f(-x) = 2x \)[/tex].
Now, let's address the second part of the problem. Given that [tex]\( x - 2k \)[/tex] is a factor of the polynomial [tex]\( 5x^3 - 10x^2k - 3x - 6 \)[/tex], we need to determine the value of [tex]\( k \)[/tex].
1. Factor: Identify that if [tex]\( x - 2k \)[/tex] is a factor, then the polynomial should be zero when [tex]\( x = 2k \)[/tex].
2. Set up the equation: Substitute [tex]\( x = 2k \)[/tex] into the polynomial:
[tex]\[
5(2k)^3 - 10(2k)^2 k - 3(2k) - 6 = 0
\][/tex]
3. Simplify:
[tex]\[
5(8k^3) - 10(4k^3) - 6k - 6 = 0
\][/tex]
[tex]\[
40k^3 - 40k^3 - 6k - 6 = 0
\][/tex]
Notice that [tex]\( 40k^3 - 40k^3 = 0 \)[/tex]:
[tex]\[
-6k - 6 = 0
\][/tex]
4. Solve for [tex]\( k \)[/tex]:
[tex]\[
-6k - 6 = 0 \implies -6k = 6 \implies k = -1
\][/tex]
Therefore, [tex]\( k = -1 \)[/tex] is the value that makes [tex]\( x - 2k \)[/tex] a factor of the given polynomial.
