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)