Find the value of k for (3x⁵ - 4x⁴ + 10x³ + kx² - 8x + 4) that when divided by (x - 2)

gives a remainder of zero.

k = - 25

Step-by-step explanation:

solving by function and roots

let f (x) = 3x⁵ - 4x⁴ + 10x³ + kx² - 8x + 4

given that when divided by (x-2), the remainder is zero, hence we can say that (x-2) is a factor of f (x), which also means that (x - 2) = 0 or x = 2 is a root of f (x)

recall if we substitute a value of x into f (x) that is a known root of a function, the functions becomes zero,

i. e substitute x = 2 into f (x)

f (2) = 3 (2) ⁵ - 4 (2) ⁴ + 10 (2) ³ + k (2) ² - 8 (2) + 4 = 0

3 (32) - 4 (16) + 10 (8) + k (4) - 8 (2) + 4 = 0

96 - 64 + 80 + 4k - 16 + 4 = 0

100 + 4k = 0

4k = - 100

k = - 25