To find the value of [tex]\( k \)[/tex] for which the polynomial [tex]\( kx^3 + 8x^2 - 4x + 10 \)[/tex] leaves a remainder of -3 when divided by [tex]\( x + 1 \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial [tex]\( f(x) \)[/tex] is divided by [tex]\( x - a \)[/tex], the remainder of this division is [tex]\( f(a) \)[/tex].
In this case, the polynomial is [tex]\( kx^3 + 8x^2 - 4x + 10 \)[/tex], and we are dividing it by [tex]\( x + 1 \)[/tex]. To use the Remainder Theorem, we need to set [tex]\( a \)[/tex] to -1 (since [tex]\( x + 1 = x - (-1) \)[/tex]).
According to the Remainder Theorem:
[tex]\[ f(-1) = -3 \][/tex]
Substitute [tex]\( -1 \)[/tex] into the polynomial:
[tex]\[ f(-1) = k(-1)^3 + 8(-1)^2 - 4(-1) + 10 \][/tex]
Simplify each term:
[tex]\[ f(-1) = k(-1) + 8(1) - 4(-1) + 10 \][/tex]
[tex]\[ f(-1) = -k + 8 + 4 + 10 \][/tex]
[tex]\[ f(-1) = -k + 22 \][/tex]
Now, set the expression equal to the remainder given by the problem:
[tex]\[ -k + 22 = -3 \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ -k + 22 = -3 \][/tex]
[tex]\[ -k = -3 - 22 \][/tex]
[tex]\[ -k = -25 \][/tex]
[tex]\[ k = 25 \][/tex]
Thus, the value of [tex]\( k \)[/tex] is:
[tex]\[ \boxed{25} \][/tex]
