Answer:
a) k = 5.
b) remainder = -52.
Explanation:
When x = 2:
4a + 2b + c + 8 = 0...... (1)
When x = 3
9a + 3b + c + 27= 0........(2)
By the remainder theorem:
f(1) = -8 so
a + b + c + 1 = -8
a + b + c + 9 = 0...........(3)
Solving the system of equations:
equation (2) - equation (1) gives:
5a + b = -19
Equation (2) - (3) gives
8a + 2b = -18
Solving the last 2 gives:
a = -10 and b = 31.
So from the first equation:
c= -4a - 2b - 8
c = -4(-10) - 2(31) - 8 = -30
So the function is:
f(x) = x^3 - 10x^2 + 31x - 30
Now as the last term is -30 , because 2 roots are 2 and 3, making 2 factors (x-2) and (x- 3) the third factor is (x - 5), so the other root is 5
a) k = 5.
When the function is divided by x + 1 , (f-1) is the remainder.
b) f(-1) = (-1)^3 - 10(-1)^2 + 31(-1) - 30
= -1 + 10 - 31 - 30
= -52.
