Factor the following high-degree polynomial completely.
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(x2+4)(x2−4)
3 left parenthesis x squared plus 4 right parenthesis left parenthesis x squared minus 4 right parenthesis
3(x4−16)
The correct answer is:
3(x^2 + 4)(x + 2)(x - 2)
To factor the polynomial 3x^4 - 48 completely, we can start by factoring out the greatest common factor, which is 3.
3(x^4 - 16)
Next, we notice that the expression inside the parentheses is a difference of squares. We can factor it accordingly:
3((x^2)^2 - 4^2)
Using the difference of squares formula, (a^2 - b^2) = (a + b)(a - b), we can factor it further:
3(x^2 - 4)(x^2 + 4)
Finally, we have factored the polynomial completely as:
3(x^2 - 4)(x^2 + 4)
To factor the polynomial 3x^4 - 48 completely, we can first find the greatest common factor (GCF) of the terms. In this case, the GCF is 3. We can factor out the GCF to rewrite the polynomial as:
3(x^4 - 16)
Now, we have a difference of squares in the parentheses, which can be factored using the formula a^2 - b^2 = (a + b)(a - b). In this case, a = x^2 and b = 4. So, we can factor the expression inside the parentheses as:
(x^2 + 4)(x^2 - 4)
The second expression, x^2 - 4, is also a difference of squares. We can use the formula again with a = x and b = 2:
(x + 2)(x - 2)
Bringing everything together, we have:
3(x^2 + 4)(x + 2)(x - 2)
Therefore, the completely factored form of the polynomial 3x^4 - 48 is 3(x^2 + 4)(x + 2)(x - 2).