Sure, let's find the value of [tex]\( k \)[/tex] such that the polynomial [tex]\( 5x^4 + 2kx^3 - 6kx^2 + 9 \)[/tex] has a remainder of 22 when divided by [tex]\( x - 1 \)[/tex].
### Step-by-Step Solution:
1. Determine the Remainder Using Polynomial Division:
When a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], the remainder is given by [tex]\( P(a) \)[/tex]. Here, [tex]\( a = 1 \)[/tex], so we need to find [tex]\( P(1) \)[/tex].
2. Evaluate the Polynomial [tex]\( P(x) = 5x^4 + 2kx^3 - 6kx^2 + 9 \)[/tex] at [tex]\( x = 1 \)[/tex]:
Let's substitute [tex]\( x = 1 \)[/tex] into the polynomial.
[tex]\[
P(1) = 5(1)^4 + 2k(1)^3 - 6k(1)^2 + 9
\][/tex]
Simplifying this,
[tex]\[
P(1) = 5 + 2k - 6k + 9
\][/tex]
Combine the like terms:
[tex]\[
P(1) = 5 + 9 + 2k - 6k
\][/tex]
Simplify further:
[tex]\[
P(1) = 14 - 4k
\][/tex]
3. Given that the remainder is 22:
Since the remainder when the polynomial is divided by [tex]\( x - 1 \)[/tex] is 22,
[tex]\[
P(1) = 22
\][/tex]
4. Set up the equation and solve for [tex]\( k \)[/tex]:
We have
[tex]\[
14 - 4k = 22
\][/tex]
Subtract 14 from both sides:
[tex]\[
-4k = 22 - 14
\][/tex]
Simplify the right-hand side:
[tex]\[
-4k = 8
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
Divide both sides by -4:
[tex]\[
k = \frac{8}{-4} = -2
\][/tex]
### Conclusion:
The value of [tex]\( k \)[/tex] that satisfies the condition is:
[tex]\[
\boxed{-2}
\][/tex]
