If `p(x)=x^(3)-3x^(2)-2x+4` , then find the value of `[p(2)+p(-2)-p(0)]` .

Options

A. `28`
B. `14`
C. `-20`
D. `16`

Answer

To find the value of \( p(2) + p(-2) - p(0) \) for the polynomial \( p(x) = x^3 - 3x^2 - 2x + 4 \), we will calculate \( p(2) \), \( p(-2) \), and \( p(0) \) step by step.

### Step 1: Calculate \( p(2) \)

Substituting \( x = 2 \) into the polynomial:

\[ p(2) = (2)^3 - 3(2)^2 - 2(2) + 4 \]

Calculating each term:

\[ = 8 - 3 \cdot 4 - 4 + 4 \]
\[ = 8 - 12 - 4 + 4 \]
\[ = 8 - 12 - 4 + 4 = 8 - 12 = -4 \]

So, \( p(2) = -4 \).

### Step 2: Calculate \( p(-2) \)

Substituting \( x = -2 \) into the polynomial:

\[ p(-2) = (-2)^3 - 3(-2)^2 - 2(-2) + 4 \]

Calculating each term:

\[ = -8 - 3 \cdot 4 + 4 + 4 \]
\[ = -8 - 12 + 4 + 4 \]
\[ = -8 - 12 + 8 = -12 \]

So, \( p(-2) = -12 \).

### Step 3: Calculate \( p(0) \)

Substituting \( x = 0 \) into the polynomial:

\[ p(0) = (0)^3 - 3(0)^2 - 2(0) + 4 \]

Calculating:

\[ = 0 - 0 - 0 + 4 = 4 \]

So, \( p(0) = 4 \).

### Step 4: Calculate \( p(2) + p(-2) - p(0) \)

Now we can substitute the values we found into the expression:

\[ p(2) + p(-2) - p(0) = -4 + (-12) - 4 \]

Calculating:

\[ = -4 - 12 - 4 \]
\[ = -16 - 4 = -20 \]

Thus, the final answer is:

\[ \boxed{-20} \]