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

Answer

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

### Step 1: Define the polynomial
Let \( f(x) = x^3 + (kx + 8)x + k \).

### Step 2: Simplify the polynomial
First, we can simplify \( f(x) \):
\[ f(x) = x^3 + kx^2 + 8x + k \]

### Step 3: Find the remainder when divided by \( x + 1 \)
Using the Remainder Theorem, the remainder when \( f(x) \) is divided by \( x + 1 \) is \( f(-1) \):
\[ f(-1) = (-1)^3 + k(-1)^2 + 8(-1) + k \]
Calculating this gives:
\[ f(-1) = -1 + k - 8 + k = 2k - 9 \]
Thus, the remainder when \( f(x) \) is divided by \( x + 1 \) is \( 2k - 9 \).

### Step 4: Find the remainder when divided by \( x - 2 \)
Now, we find the remainder when \( f(x) \) is divided by \( x - 2 \), which is \( f(2) \):
\[ f(2) = (2)^3 + k(2)^2 + 8(2) + k \]
Calculating this gives:
\[ f(2) = 8 + 4k + 16 + k = 5k + 24 \]
Thus, the remainder when \( f(x) \) is divided by \( x - 2 \) is \( 5k + 24 \).

### Step 5: Set up the equation based on the problem statement
According to the problem, the sum of the two remainders is 1:
\[ (2k - 9) + (5k + 24) = 1 \]
Simplifying this equation:
\[ 2k - 9 + 5k + 24 = 1 \]
\[ 7k + 15 = 1 \]
\[ 7k = 1 - 15 \]
\[ 7k = -14 \]
\[ k = -2 \]

### Conclusion
The value of \( k \) is \( -2 \).

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