If `f(x)=x^(3)-x^(2)`, then `f(x+1)` is :

Options

A. `x^(3)-x^(2)+1`
B. `x(x+1)^(2)`
C. `x^(2)(x-1)`
D. none of these

Answer

To find \( f(x+1) \) for the function \( f(x) = x^3 - x^2 \), we will follow these steps:

### Step 1: Write down the function
The given function is:
\[ f(x) = x^3 - x^2 \]

### Step 2: Substitute \( x+1 \) into the function
To find \( f(x+1) \), we need to replace \( x \) in the function with \( x+1 \):
\[ f(x+1) = (x+1)^3 - (x+1)^2 \]

### Step 3: Expand \( (x+1)^3 \)
Using the binomial expansion:
\[ (x+1)^3 = x^3 + 3x^2 + 3x + 1 \]

### Step 4: Expand \( (x+1)^2 \)
Again using the binomial expansion:
\[ (x+1)^2 = x^2 + 2x + 1 \]

### Step 5: Substitute the expansions back into the function
Now, we substitute the expansions back into the equation for \( f(x+1) \):
\[ f(x+1) = (x^3 + 3x^2 + 3x + 1) - (x^2 + 2x + 1) \]

### Step 6: Simplify the expression
Now, we simplify the expression:
\[ f(x+1) = x^3 + 3x^2 + 3x + 1 - x^2 - 2x - 1 \]
Combine like terms:
\[ f(x+1) = x^3 + (3x^2 - x^2) + (3x - 2x) + (1 - 1) \]
This simplifies to:
\[ f(x+1) = x^3 + 2x^2 + x \]

### Final Answer
Thus, the final result for \( f(x+1) \) is:
\[ f(x+1) = x^3 + 2x^2 + x \]

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