Factor the following high-degree polynomial completely.
48x6−3x2
(1 point)
Responses
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)
3 x squared left parenthesis 4 x squared plus 1 right parenthesis left parenthesis 2 x plus 1 right parenthesis left parenthesis 2 x 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(8x2+1)(8x2−1)
3x^2(8x^2+1)(8x^2-1)
To factor the polynomial 48x^6 - 3x^2 completely, we first notice that both terms have a common factor of 3x^2. Thus, we can factor it out:
3x^2(16x^4 - 1)
Now, we can rewrite the expression inside the parentheses as a difference of squares:
3x^2((4x^2)^2 - 1^2)
Applying the formula for the difference of squares (a^2 - b^2 = (a + b)(a - b)), we get:
3x^2(4x^2 + 1)(4x^2 - 1)
Now, we have factored the polynomial completely as:
3x^2(4x^2 + 1)(4x^2 - 1)
To factor the polynomial 48x^6 - 3x^2 completely, we can start by looking for any common factors among the terms. In this case, both terms have a common factor of 3x^2. So, we can factor out 3x^2 from both terms:
3x^2(16x^4 - 1)
Next, we can observe that the expression inside the parentheses, 16x^4 - 1, is a difference of squares. It can be factored as (4x^2)^2 - 1^2:
3x^2((4x^2 - 1)(4x^2 + 1))
Now, we have factored the polynomial completely as 3x^2(4x^2 - 1)(4x^2 + 1). This is the final factored form.