Factor the following high-degree polynomial completely. 48x6−3x2

Question

(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)(4x + 1)(4x − 1)
3x2(4x2+1)(4x + 1)(4x − 1)

3x2(4x2+1)(2x + 1)(2x − 1)
3x2(4x2+1)(2x + 1)(2x − 1)

3x2(4x2+1)(4x2 − 1)

Answer 1

To factor the polynomial \( 48x^6 - 3x^2 \), we start by looking for the greatest common factor (GCF) of the two terms:

1. The GCF is \( 3x^2 \).
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)