Factors of `3m ^(5) ` - 48 m are

Options

A. `3m ( m -1) ( m - 3)`
B. `3m ( m -2) ( m +2) ( m ^(2) + 4)`
C. `3m ( m -1) ( m -2) ( m+1)`
D. `m (m-1) (m +2) ( m^(2) + 4)`

Answer

To factor the expression \(3m^5 - 48m\), we will follow these steps:

### Step 1: Identify the common factor
First, we need to find the common factor in both terms of the expression \(3m^5\) and \(-48m\). Both terms have a common factor of \(3m\).

### Step 2: Factor out the common factor
Now, we can factor out \(3m\) from the expression:
\[ 3m^5 - 48m = 3m(m^4 - 16) \]

### Step 3: Recognize the difference of squares
The expression \(m^4 - 16\) can be recognized as a difference of squares since \(16\) is \(4^2\). We can rewrite it as:
\[ m^4 - 4^2 \]

### Step 4: Apply the difference of squares formula
Using the difference of squares formula \(a^2 - b^2 = (a + b)(a - b)\), we can factor \(m^4 - 4^2\):
\[ m^4 - 4^2 = (m^2 + 4)(m^2 - 4) \]

### Step 5: Factor \(m^2 - 4\) further
Notice that \(m^2 - 4\) is also a difference of squares, where \(4\) is \(2^2\). We can factor it further:
\[ m^2 - 4 = (m + 2)(m - 2) \]

### Step 6: Combine all factors
Now, we can combine all the factors we have found:
\[ 3m(m^2 + 4)(m + 2)(m - 2) \]

### Final Answer
Thus, the complete factorization of \(3m^5 - 48m\) is:
\[ 3m(m^2 + 4)(m + 2)(m - 2) \]

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