When 2x^3 + x^2 - 2kx + f is divided by x-1, the remainder is -4, and when it is divided by x+2, the remainder is 11. Determine the values of k and f.

Answer 1

do long division, divide by x-1 = 2x^2 +3x + 3-2k with remainder f+3-2k =-4

f-2k = -7

divide by x+2 gives 2x^2 -3x +6 -2k with remainder f-12+4k = 11

f+4k = 23

f+4k=23

f-2k-7 subtract to get

6k =30

k=5

plug that into either equation

to solve for f

f=-7+2(5) = 3

f=3

k=5, f=3