)Find a value of s such that x – 4 is a factor of f(x) = x3 - 2x2+ sx + 4.

division reveals that the remainder is 4s+36

so, if s = -9, the remainder is zero.

check:
x^3 - 2x^2 - 9x + 4 = (x-4)(x^2 + 2x - 1)

To find a value of s such that x - 4 is a factor of f(x), we can use the Remainder Theorem. According to the theorem, if x - a is a factor of a polynomial f(x), then f(a) = 0.

In this case, since x - 4 is a factor of f(x), it means that if we substitute x = 4 into f(x), we should get 0 as the result. Let's calculate f(4) and set it equal to 0 to solve for the value of s.

Substituting x = 4 into f(x), we have:
f(4) = (4)^3 - 2(4)^2 + s(4) + 4
= 64 - 32 + 4s + 4
= 4s + 36

Setting f(4) equal to 0, we get:
4s + 36 = 0

To solve for s, we isolate the variable:
4s = -36
s = -36/4
s = -9

Therefore, the value of s that makes x - 4 a factor of f(x) = x^3 - 2x^2 + sx + 4 is s = -9.