F(x) = x3 + ax2 + bx + c The roots of f(x) = 0 are 2, 3, and k.
When f(x) is divided by x - 1, the remainder is -8.
a) Find the value of k.
b) Find the remainder when f(x) is divided by x + 1.​

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.

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