Factorize the polynomial `-r^(2)+p^(2)+q^(2)-2pq`.

Options

A. `(p-q-r)(p-q+r)`
B. `(p+q+r)(p-q-r)`
C. `(p-q)(q-r)`
D. `(p-r)(q-r)`

Answer

To factorize the polynomial \(-r^2 + p^2 + q^2 - 2pq\), we can follow these steps:

### Step 1: Rearrange the terms
Rearranging the polynomial gives us:
\[ -p^2 + q^2 - 2pq + p^2 \]
This can be rewritten as:
\[ -p^2 + (q^2 - 2pq + p^2) \]

### Step 2: Recognize a perfect square
Notice that \(q^2 - 2pq + p^2\) can be recognized as a perfect square:
\[ q^2 - 2pq + p^2 = (q - p)^2 \]
So, we can rewrite the polynomial as:
\[ -(r^2) + (q - p)^2 \]

### Step 3: Apply the difference of squares formula
We can now apply the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\). Here, we can let \(a = (q - p)\) and \(b = r\):
\[ -(r^2) + (q - p)^2 = (q - p)^2 - r^2 \]
This can be factored as:
\[ ((q - p) + r)((q - p) - r) \]

### Step 4: Write the final factorized form
Thus, the factorized form of the polynomial \(-r^2 + p^2 + q^2 - 2pq\) is:
\[ (q - p + r)(q - p - r) \]

### Final Answer:
The factorized form is:
\[ (q - p + r)(q - p - r) \]

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