Factor the following high-degree polynomial completely.

Question

48x^6 - 3x^2

A
3x2(4x2+1)(4x+1)(4x−1)3x^2\left(4x^2+1\right)\left(4x+1\right)\left(4x-1\right)3x
2
(4x
2
+1)(4x+1)(4x−1)

B
3x2(4x2+1)(4x2−1)3x^2\left(4x^2+1\right)\left(4x^2-1\right)3x
2
(4x
2
+1)(4x
2
−1)

C
3x2(8x2+1)(8x2−1)3x^2\left(8x^2+1\right)\left(8x^2-1\right)3x
2
(8x
2
+1)(8x
2
−1)

D
3x2(4x2+1)(2x+1)(2x−1)3x^2\left(4x^2+1\right)\left(2x+1\right)\left(2x-1\right)3x
2
(4x
2
+1)(2x+1)(2x−1)

Answer 1

To factor the polynomial \(48x^6 - 3x^2\) completely, we first look for common factors.

1. Factor out the greatest common factor (GCF): The GCF of the terms \(48x^6\) and \(-3x^2\) is \(3x^2\). \[ 48x^6 - 3x^2 = 3x^2(16x^4 - 1) \]

2. Factor the difference of squares: Now, we need to factor \(16x^4 - 1\). Recognizing this is a difference of squares: \[ 16x^4 - 1 = (4x^2)^2 - 1^2 = (4x^2 - 1)(4x^2 + 1) \]

3. Further factor \(4x^2 - 1\): The term \(4x^2 - 1\) can also be factored as: \[ 4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1) \]

4. Combining the factors: Putting it all together, we have: \[ 48x^6 - 3x^2 = 3x^2(4x^2 - 1)(4x^2 + 1) = 3x^2(2x - 1)(2x + 1)(4x^2 + 1) \]

Thus, the completely factored form of \(48x^6 - 3x^2\) is: \[ 3x^2(2x - 1)(2x + 1)(4x^2 + 1) \]

Looking at the options you provided:

The correct answer is D: \[ 3x^2(4x^2+1)(2x+1)(2x−1) \]