Question

select the correct answer.

which expression is equivalent to this polynomial?

(4x^2 + 27)

\\((2x + 3\sqrt{3}i)(2x - 3\sqrt{3}i)\\)
\\((2x + 9i)(2x - 3i)\\)
\\((2x + 3\sqrt{3})^2\\)
\\((2x + 3\sqrt{3})(2x - 3\sqrt{3})\\)

Explanation:

Rewrite the expression as a difference of squares

$$ 4x^2 + 27 = (2x)^2 - (-27) $$

Express the constant term using the imaginary unit

$$ -27 = (i\sqrt{27})^2 = (3\sqrt{3}i)^2 $$

Apply the difference of squares identity

$$ (2x)^2 - (3\sqrt{3}i)^2 = (2x + 3\sqrt{3}i)(2x - 3\sqrt{3}i) $$

Answer:

• (A) \((2x + 3\sqrt{3}i)(2x - 3\sqrt{3}i)\) (Correct answer)
• (B) \((2x + 9i)(2x - 3i)\)
• (C) \((2x + 3\sqrt{3})^2\)
• (D) \((2x + 3\sqrt{3})(2x - 3\sqrt{3})\)