Factor the following high-degree polynomial completely.
48x^6−3x^2
3x^2(4x^2+1)(2x+1)(2x-1)
3x^2(8x^2+1)(8x^2-1)
3x^2(4x^2+1)(4x+1)(4x-1)
3x^2(4x^2+1)(4x^2-1)
The correct factorization is: 3x^2(4x^2+1)(2x+1)(2x-1)
To factor the polynomial 48x^6 - 3x^2 completely, we can start by factoring out the greatest common factor, which is 3x^2. This leaves us with:
3x^2(16x^4 - 1)
Next, we can notice that the expression in parentheses is a difference of squares. By using the formula a^2 - b^2 = (a + b)(a - b), we can factor it further:
3x^2((4x^2)^2 - 1^2)
3x^2(4x^2 + 1)(4x^2 - 1)
Finally, we can apply the difference of squares formula again to the term (4x^2 - 1), which gives us:
3x^2(4x^2 + 1)(2x + 1)(2x - 1)
Therefore, the given polynomial completely factors to: 3x^2(4x^2 + 1)(2x + 1)(2x - 1).