Factor completely, or state that the polynomial is prime.
\[
27-12 x^{2}
\]
Select the correct choice below and fill in any answer boxes within your choice.
A. $27-12 x^{2}=$
B. The polynomial is prime.
Answer
The factored form of the polynomial \(27 - 12x^2\) is \(\boxed{(3\sqrt{3} - 2x\sqrt{3})(3\sqrt{3} + 2x\sqrt{3})}\)
Step 1 ：The given polynomial is a quadratic in the form of \(a^2 - b^2\), which can be factored using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\).
Step 2 ：Here, \(a = \sqrt{27} = 3\sqrt{3}\) and \(b = \sqrt{12x^2} = 2x\sqrt{3}\).
Step 3 ：So, we can factor the polynomial as \((3\sqrt{3} - 2x\sqrt{3})(3\sqrt{3} + 2x\sqrt{3})\).
Step 4 ：The factored form of the polynomial \(27 - 12x^2\) is \(\boxed{(3\sqrt{3} - 2x\sqrt{3})(3\sqrt{3} + 2x\sqrt{3})}\)
