Factor the following high-degree polynomial completely.

Question

3x4−48

(1 point)
Responses

3(x2+4)(x−2)(x+2)
3 left parenthesis x squared plus 4 right parenthesis left parenthesis x minus 2 right parenthesis left parenthesis x plus 2 right parenthesis

3(x2−8)(x2+8)
3 left parenthesis x squared minus 8 right parenthesis left parenthesis x squared plus 8 right parenthesis

3(x4−16)
3 left parenthesis x superscript 4 baseline minus 16 right parenthesis

3(x2+4)(x2−4)

Answer 1

To factor the polynomial \(3x^4 - 48\) completely, we can follow these steps:

1. Factor out the greatest common factor (GCF) from the polynomial.

2. Rewrite the expression:

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

3. Now, notice that \(x^4 - 16\) is a difference of squares:

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

4. The term \(x^2 - 4\) can be further factored again since it, too, is a difference of squares:

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

5. Therefore, we can rewrite our factorization:

   \[ 3(x^4 - 16) = 3((x - 2)(x + 2))(x^2 + 4) \]

Putting all of this together, the completely factored form of \(3x^4 - 48\) is:

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

So, the correct response would be:

3(x² - 8)(x² + 8) is not a valid option for \(3(x^4 - 16)\) here, so the answer is:

3(x² + 4)(x - 2)(x + 2)