If `f(x) =5x^3 +4x^2 - 13 x-25` and ` f(x-3) = 5x^3 - 41 x^2 + 98 x + k ` then ` k=`

Options

A. 85
B. `-85`
C. `105`
D. `-105`

Answer

To find the value of \( k \) in the given equations, we will follow these steps:

### Step 1: Write down the functions
We have two functions:
1. \( f(x) = 5x^3 + 4x^2 - 13x - 25 \)
2. \( f(x-3) = 5x^3 - 41x^2 + 98x + k \)

### Step 2: Set up the equation for \( k \)
From the problem statement, we know that:
\[ k = f(-3) \]
This means we need to calculate \( f(-3) \) using the first function.

### Step 3: Calculate \( f(-3) \)
Substituting \( x = -3 \) into \( f(x) \):
\[ f(-3) = 5(-3)^3 + 4(-3)^2 - 13(-3) - 25 \]

### Step 4: Simplify the expression
Calculating each term:
- \( (-3)^3 = -27 \) so \( 5(-27) = -135 \)
- \( (-3)^2 = 9 \) so \( 4(9) = 36 \)
- \( -13(-3) = 39 \)
- The constant term is \( -25 \)

Now, substituting these values back into the equation:
\[ f(-3) = -135 + 36 + 39 - 25 \]

### Step 5: Combine the terms
Now, we will combine the terms step by step:
1. Combine \( -135 + 36 = -99 \)
2. Then, \( -99 + 39 = -60 \)
3. Finally, \( -60 - 25 = -85 \)

Thus, we find:
\[ f(-3) = -85 \]

### Step 6: Substitute back to find \( k \)
Since \( k = f(-3) \):
\[ k = -85 \]

### Final Answer
The value of \( k \) is:
\[ \boxed{-85} \]

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