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Bot GPT 4-o mini To factor the polynomial \(48x^6 - 3x^2\) completely, we start by factoring out the greatest common factor:
1. Identify the GCF of the terms \(48x^6\) and \(-3x^2\). The GCF is \(3x^2\).
\[ 48x^6 - 3x^2 = 3x^2(16x^4 - 1) \]
2. Now we need to factor the expression \(16x^4 - 1\). This is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\):
\[ 16x^4 - 1 = (4x^2)^2 - (1)^2 = (4x^2 - 1)(4x^2 + 1) \]
3. Next, we can further factor \(4x^2 - 1\) since it is also a difference of squares:
\[ 4x^2 - 1 = (2x)^2 - (1)^2 = (2x - 1)(2x + 1) \]
4. The expression \(4x^2 + 1\) does not factor further over the real numbers.
Putting it all together, we have:
\[
48x^6 - 3x^2 = 3x^2(4x^2 - 1)(4x^2 + 1) = 3x^2(2x - 1)(2x + 1)(4x^2 + 1)
\]
Therefore, the fully factored form of \(48x^6 - 3x^2\) is:
\[
3x^2(2x - 1)(2x + 1)(4x^2 + 1)
\]
Among the provided responses, the one that corresponds to this factorization is:
**3x²(4x² + 1)(4x - 1)(4x + 1)** would be incorrect due to \(4x^2+1\) staying as is and \(2x-1\), \(2x+1\) should be included.
Hence, none of the provided responses were completely accurate. However, the correct approach implies taking \(48x^6 - 3x^2\) and breaking it to \(3x^2(4x^4-1)\) and then \(3x^2(2x-1)(2x+1)(4x^2+1)\) as the breakdown.