Question

Factor the following 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)(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)(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)(4x+1)(4x−1)

Answer 1

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

The GCF of \(48x^6\) and \(-3x^2\) is \(3x^2\). Factoring that out, we have:

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

Next, we notice that \(16x^4 - 1\) is a difference of squares:

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

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

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

Now, we can substitute everything back into our expression:

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

So the complete factorization of the polynomial \(48x^6 - 3x^2\) is:

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

However, the given options don't exactly match this factorization format. The closest option in the provided responses that matches our factorization is:

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

b) \(3 x^2(4 x^2 + 1)(2 x + 1)(2 x - 1)\)