Find the value of 'k' if (x-2) is a factor of `x^3+2x^2-kx+10`. Hence determine whether (x+5) is also a factor.

Answer

To solve the problem, we need to find the value of 'k' such that (x - 2) is a factor of the polynomial \( f(x) = x^3 + 2x^2 - kx + 10 \). Then, we will check if (x + 5) is also a factor of the polynomial.

### Step 1: Set up the polynomial
Let \( f(x) = x^3 + 2x^2 - kx + 10 \).

### Step 2: Use the Factor Theorem
Since (x - 2) is a factor of \( f(x) \), we can use the Factor Theorem, which states that if (x - a) is a factor of a polynomial, then \( f(a) = 0 \). Here, \( a = 2 \).

### Step 3: Substitute x = 2 into the polynomial
We substitute \( x = 2 \) into \( f(x) \):
\[ f(2) = 2^3 + 2(2^2) - k(2) + 10 \]
Calculating this gives:
\[ f(2) = 8 + 2(4) - 2k + 10 \]
\[ f(2) = 8 + 8 - 2k + 10 \]
\[ f(2) = 26 - 2k \]

### Step 4: Set the equation to zero
Since \( f(2) = 0 \) (because (x - 2) is a factor), we have:
\[ 26 - 2k = 0 \]

### Step 5: Solve for k
Rearranging the equation gives:
\[ 2k = 26 \]
\[ k = \frac{26}{2} = 13 \]

### Step 6: Check if (x + 5) is a factor
Now we need to check if (x + 5) is also a factor. We set \( x + 5 = 0 \) which gives \( x = -5 \).

### Step 7: Substitute x = -5 into the polynomial
Now substitute \( x = -5 \) into \( f(x) \):
\[ f(-5) = (-5)^3 + 2(-5)^2 - 13(-5) + 10 \]
Calculating this gives:
\[ f(-5) = -125 + 2(25) + 65 + 10 \]
\[ f(-5) = -125 + 50 + 65 + 10 \]
\[ f(-5) = -125 + 125 = 0 \]

### Conclusion
Since \( f(-5) = 0 \), we conclude that (x + 5) is also a factor of the polynomial.

### Final Answer
The value of \( k \) is \( 13 \), and both (x - 2) and (x + 5) are factors of the polynomial \( x^3 + 2x^2 - 13x + 10 \).