Factor the following high-degree polynomial completely.

Question

48x6−3x2

(1 point)
Responses

3x2(8x2+1)(8x2−1)
3 x squared left parenthesis 8 x squared plus 1 right parenthesis left parenthesis 8 x squared minus 1 right parenthesis

3x2(4x2+1)(4x+1)(4x−1)
3 x squared left parenthesis 4 x squared plus 1 right parenthesis left parenthesis 4 x plus 1 right parenthesis left parenthesis 4 x minus 1 right parenthesis

3x2(4x2+1)(4x2−1)
3 x squared left parenthesis 4 x squared plus 1 right parenthesis left parenthesis 4 x squared minus 1 right parenthesis

3x2(4x2+1)(2x+1)(2x−1)

GPT-4o mini · Answer

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.