Factor the following high-degree polynomial completely. 48x6−3x2 (1 point) Responses 3x2(8x2+1)(8x2−1) 3 x squared open paren 8 x squared plus 1 close paren open paren 8 x squared minus 1 close paren 3x2(4x2+1)(4x2−1) 3 x squared open paren 4 x squared plus 1 close paren open paren 4 x squared minus 1 close paren 3x2(4x2+1)(2x+1)(2x−1) 3 x squared open paren 4 x squared plus 1 close paren open paren 2 x plus 1 close paren open paren 2 x minus 1 close paren 3x2(4x2+1)(4x+1)(4x−1)

Answer 1

To factor the polynomial \( 48x^6 - 3x^2 \) completely, we start by factoring out the greatest common factor (GCF), which is \( 3x^2 \):

\[ 48x^6 - 3x^2 = 3x^2(16x^4 - 1) \]

Next, we observe that \( 16x^4 - 1 \) is a difference of squares, which can be factored as follows:

\[ 16x^4 - 1 = (4x^2)^2 - 1^2 = (4x^2 - 1)(4x^2 + 1) \]

Now, we have factored \( 16x^4 - 1 \) into \( (4x^2 - 1)(4x^2 + 1) \).

The \( 4x^2 - 1 \) term can also be factored further as it is another difference of squares:

\[ 4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1) \]

Putting all of this 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 the polynomial \( 48x^6 - 3x^2 \) is:

\[ 3x^2(2x - 1)(2x + 1)(4x^2 + 1) \]

Now, looking at the options provided, none of them completely match our final factorization:

1. \( 3x^2(8x^2+1)(8x^2−1) \)
2. \( 3x^2(4x^2+1)(4x^2−1) \)
3. \( 3x^2(4x^2+1)(2x+1)(2x−1) \)
4. \( 3x^2(4x^2+1)(4x+1)(4x−1) \)

However, option 3 actually contains the correct factors, but lists \( (4x^2 + 1) \) separately along with the fully factored \( (2x - 1)(2x + 1) \)—it merely does not display \( (4x^2 - 1) \) explicitly, since this factor has split into two linear factors \( (2x - 1) \) and \( (2x + 1) \).

Therefore, the correct factorization in response form is:

\[ \boxed{3x^2(4x^2 + 1)(2x + 1)(2x - 1)} \]