Using the Remainder Theorem find the remainders obtained when `x^(3)+(kx+8)x+k` is divided by `x-1andx-2`.
Hence find k if the sum of the remainders is 1.

Answer

To solve the problem using the Remainder Theorem, we will follow these steps:

### Step 1: Define the polynomial
Let \( P(x) = x^3 + (kx + 8)x + k \).
We can rewrite this as:
\[ P(x) = x^3 + kx^2 + 8x + k \]

### Step 2: Find the remainder when divided by \( x - 1 \)
According to the Remainder Theorem, the remainder \( r_1 \) when \( P(x) \) is divided by \( x - 1 \) is given by \( P(1) \):
\[ P(1) = 1^3 + k(1^2) + 8(1) + k = 1 + k + 8 + k = 2k + 9 \]
So, \( r_1 = 2k + 9 \).

### Step 3: Find the remainder when divided by \( x - 2 \)
Similarly, the remainder \( r_2 \) when \( P(x) \) is divided by \( x - 2 \) is given by \( P(2) \):
\[ P(2) = 2^3 + k(2^2) + 8(2) + k = 8 + 4k + 16 + k = 5k + 24 \]
So, \( r_2 = 5k + 24 \).

### Step 4: Set up the equation for the sum of the remainders
According to the problem, the sum of the remainders is given to be 1:
\[ r_1 + r_2 = 1 \]
Substituting the values of \( r_1 \) and \( r_2 \):
\[ (2k + 9) + (5k + 24) = 1 \]
This simplifies to:
\[ 7k + 33 = 1 \]

### Step 5: Solve for \( k \)
Now, we will solve for \( k \):
\[ 7k = 1 - 33 \]
\[ 7k = -32 \]
\[ k = -\frac{32}{7} \]

### Final Answer
Thus, the value of \( k \) is:
\[ k = -\frac{32}{7} \]