Among the expression `(1 - 3p), [1 - (3P)^(2)], [1 - (3P)^(3)]` and `[1 - (3P)^(4)]` the number of factors of `(1 - 81 P^4)` is :

Options

A. one
B. two
C. four
D. three

Answer

To find the number of factors of the expression \(1 - 81p^4\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 1 - 81p^4 \]
We can observe that \(81\) is \(3^4\), so we can rewrite the expression as:
\[ 1 - (3p^2)^4 \]

### Step 2: Apply the difference of squares
We can use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\). Here, we can let:
\[ a = 1 \quad \text{and} \quad b = (3p^2)^2 \]
Thus, we have:
\[ 1 - (3p^2)^4 = (1 - (3p^2)^2)(1 + (3p^2)^2) \]

### Step 3: Simplify the first factor
Next, we simplify the first factor:
\[ 1 - (3p^2)^2 = 1 - 9p^4 \]
Again, we can apply the difference of squares:
\[ 1 - 9p^4 = (1 - 3p^2)(1 + 3p^2) \]

### Step 4: Combine the factors
Now, we can combine the factors we have found so far:
\[ 1 - 81p^4 = (1 - 3p^2)(1 + 3p^2)(1 + 9p^4) \]

### Step 5: Factor the remaining expression
Now, we can factor \(1 - 3p^2\) further using the difference of squares:
\[ 1 - 3p^2 = (1 - \sqrt{3}p)(1 + \sqrt{3}p) \]
Thus, we have:
\[ 1 - 81p^4 = (1 - 3p)(1 + 3p)(1 + 3p^2)(1 + 9p^4) \]

### Step 6: Count the factors
Now we can count the distinct factors:
1. \(1 - 3p\)
2. \(1 + 3p\)
3. \(1 + 3p^2\)
4. \(1 + 9p^4\)

In total, we have **4 distinct factors**.

### Final Answer
The number of factors of \(1 - 81p^4\) is **4**.

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