Question

Factor the following high-degree polynomial completely.
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 \( 48x^6 - 3x^2 \), let's start with factoring out the greatest common factor (GCF).

1. **Identify the GCF**: The GCF of \( 48x^6 \) and \( -3x^2 \) is \( 3x^2 \).

2. **Factor out the GCF**:

\[
48x^6 - 3x^2 = 3x^2(16x^4 - 1)
\]

3. **Factor the remaining expression**: The expression \( 16x^4 - 1 \) is a difference of squares, which can be factored as follows:

\[
16x^4 - 1 = (4x^2)^2 - 1^2 = (4x^2 - 1)(4x^2 + 1)
\]

4. **Factor \( 4x^2 - 1 \)**: The term \( 4x^2 - 1 \) is also a difference of squares:

\[
4x^2 - 1 = (2x)^2 - 1^2 = (2x - 1)(2x + 1)
\]

5. **Compile all factors**: Now we can write the fully factored form:

\[
48x^6 - 3x^2 = 3x^2(4x^2 + 1)(2x - 1)(2x + 1)
\]

Thus, the completely factored form of the polynomial \( 48x^6 - 3x^2 \) is:

\[
3x^2(4x^2 + 1)(2x - 1)(2x + 1)
\]

From the given options, it matches with:

**D**: \( 3x^2(4x^2 + 1)(2x + 1)(2x - 1) \)

Therefore, the answer is **D**.