Factor the polynomial completely: 48x^6−3x^2 (2 points)
1. 3x^2(4x^2+1)(4x^2−1)
2. 3x^2(2x^2+1)(2x^2−1)
3. 3x^2(4x^2+1)(2x+1)(2x−1)
4. 3x^2(4x^2+1)(4x+1)(4x−1)
To factor the polynomial 48x^6 - 3x^2 completely, we can start by factoring out the greatest common factor, which is 3x^2.
Step 1: Factor out 3x^2:
3x^2(16x^4 - 1)
Next, we can observe that the expression inside the parentheses is a difference of squares.
Step 2: Apply the difference of squares formula:
3x^2(4x^2 + 1)(4x^2 - 1)
The expression (4x^2 - 1) is a difference of squares, so we can further factor it.
Step 3: Apply the difference of squares formula again:
3x^2(4x^2 + 1)(2x + 1)(2x - 1)
Therefore, the polynomial 48x^6 - 3x^2 is factored completely as 3x^2(4x^2 + 1)(2x + 1)(2x - 1).
The correct answer is option 3: 3x^2(4x^2 + 1)(2x + 1)(2x - 1).
The correct answer is 2. 3x^2(2x^2+1)(2x^2−1)
To factor the polynomial 48x^6 - 3x^2 completely, we need to look for any common factors and then use factoring techniques.
Step 1: Find the greatest common factor (GCF) of the terms 48x^6 and -3x^2. In this case, the GCF is 3x^2.
Step 2: Divide each term by the GCF: (48x^6)/(3x^2) - (3x^2)/(3x^2). Simplifying, we get 16x^4 - 1.
Now we need to factor the expression 16x^4 - 1. This can be done as the difference of squares.
Step 3: Apply the difference of squares identity, which states that a^2 - b^2 = (a + b)(a - b). In our case, a = 4x^2 and b = 1.
Using this identity, we can rewrite 16x^4 - 1 as (4x^2 + 1)(4x^2 - 1).
The final factored form of the given polynomial is: 3x^2(4x^2 + 1)(4x^2 - 1).
So, the correct answer is option 1: 3x^2(4x^2 + 1)(4x^2 - 1).