To solve this problem, we need to find the value of [tex]\( k \)[/tex] such that the remainders of the polynomials [tex]\( kx^3 + 3x^2 - 3 \)[/tex] and [tex]\( 2x^3 - 5x + k \)[/tex] when divided by [tex]\( x-4 \)[/tex] are the same.
### Step-by-Step Solution:
1. Substitute [tex]\( x = 4 \)[/tex] into both polynomials and set their remainders equal:
- When a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], the remainder is given by [tex]\( f(a) \)[/tex].
2. Calculate the remainder of [tex]\( kx^3 + 3x^2 - 3 \)[/tex] when [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = k(4)^3 + 3(4)^2 - 3 = k \cdot 64 + 3 \cdot 16 - 3
\][/tex]
Simplify each term:
[tex]\[
f(4) = 64k + 48 - 3
\][/tex]
Combine the constants:
[tex]\[
f(4) = 64k + 45
\][/tex]
3. Calculate the remainder of [tex]\( 2x^3 - 5x + k \)[/tex] when [tex]\( x = 4 \)[/tex]:
[tex]\[
g(4) = 2(4)^3 - 5(4) + k = 2 \cdot 64 - 5 \cdot 4 + k
\][/tex]
Simplify each term:
[tex]\[
g(4) = 128 - 20 + k
\][/tex]
Combine the constants:
[tex]\[
g(4) = 108 + k
\][/tex]
4. Set the remainders equal to each other and solve for [tex]\( k \)[/tex]:
We have:
[tex]\[
64k + 45 = 108 + k
\][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[
64k - k = 108 - 45
\][/tex]
Combine like terms:
[tex]\[
63k = 63
\][/tex]
Divide both sides by 63:
[tex]\[
k = 1
\][/tex]
### Conclusion
Thus, the value of [tex]\( k \)[/tex] that ensures the remainders are the same for both polynomials when divided by [tex]\( x-4 \)[/tex] is:
[tex]\[
\boxed{1}
\][/tex]
