To factor the polynomial \( 48x^6 - 3x^2 \), we start by looking for the greatest common factor (GCF) of the two terms:
- The GCF is \( 3x^2 \).
- Factoring out \( 3x^2 \) gives us:
\[ 48x^6 - 3x^2 = 3x^2(16x^4 - 1) \]
Next, we need to factor the expression inside the parentheses, \( 16x^4 - 1 \). This expression is a difference of squares:
\[ 16x^4 - 1 = (4x^2)^2 - (1)^2 = (4x^2 - 1)(4x^2 + 1) \]
Now, we can factor \( 4x^2 - 1 \) further, since it's also a difference of squares:
\[ 4x^2 - 1 = (2x)^2 - (1)^2 = (2x - 1)(2x + 1) \]
Putting everything together, we have:
\[ 48x^6 - 3x^2 = 3x^2(4x^2 - 1)(4x^2 + 1) = 3x^2(2x - 1)(2x + 1)(4x^2 + 1) \]
Thus, the complete factorization of \( 48x^6 - 3x^2 \) is:
\[ 3x^2(2x - 1)(2x + 1)(4x^2 + 1) \]
Looking at your responses, the closest match would be:
3x²(4x² + 1)(2x + 1)(2x - 1)