Find the zeroes of the polynomial` p(x)=(x-2)^(2)-(x+2)^(2)`.

Options

A. 0
B. 2
C. -2
D. 1

Explanation

To find the zeros of the polynomial \( p(x) = (x - 2)^2 - (x + 2)^2 \), we will follow these steps:

### Step 1: Expand the polynomial
We start by expanding both squares in the expression.

\[ p(x) = (x - 2)^2 - (x + 2)^2 \]

Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):

\[ (x - 2)^2 = x^2 - 4x + 4 \]
\[ (x + 2)^2 = x^2 + 4x + 4 \]

Now substituting these expansions back into the polynomial:

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

### Step 2: Simplify the expression
Now, we simplify the expression by distributing the negative sign and combining like terms:

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

Notice that \( x^2 \) cancels out:

\[ p(x) = -4x - 4x + 4 - 4 \]
\[ p(x) = -8x \]

### Step 3: Set the polynomial equal to zero
To find the zeros of the polynomial, we set \( p(x) \) equal to zero:

\[ -8x = 0 \]

### Step 4: Solve for \( x \)
Now, we solve for \( x \):

\[ x = 0 \]

### Conclusion
Thus, the zero of the polynomial \( p(x) = (x - 2)^2 - (x + 2)^2 \) is:

\[ \text{Zero: } x = 0 \]

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