If ` ( x - 4) ( x^(2) + 4 x + 16 ) = x^(3) - p ` then p is equal to

Options

A. A)27
B. B)8
C. C)64
D. D)0

Answer

To solve the equation \( (x - 4)(x^2 + 4x + 16) = x^3 - p \), we will follow these steps:

### Step 1: Expand the left-hand side
We need to expand the expression \( (x - 4)(x^2 + 4x + 16) \).

Using the distributive property (also known as the FOIL method for binomials), we have:
\[ (x - 4)(x^2 + 4x + 16) = x(x^2 + 4x + 16) - 4(x^2 + 4x + 16) \]

Now, let's calculate each part:
1. \( x(x^2 + 4x + 16) = x^3 + 4x^2 + 16x \)
2. \( -4(x^2 + 4x + 16) = -4x^2 - 16x - 64 \)

Now, combine these results:
\[ x^3 + 4x^2 + 16x - 4x^2 - 16x + 64 \]

### Step 2: Combine like terms
Now, we combine the like terms:
\[ x^3 + (4x^2 - 4x^2) + (16x - 16x) - 64 = x^3 + 0 + 0 - 64 = x^3 - 64 \]

### Step 3: Set the equation equal to the right-hand side
Now we have:
\[ x^3 - 64 = x^3 - p \]

### Step 4: Solve for \( p \)
By comparing both sides of the equation, we see that:
\[ -p = -64 \]

Thus, we find:
\[ p = 64 \]

### Final Answer
The value of \( p \) is \( 64 \).

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