Question

which expression is equivalent to this polynomial?
$4x^2 + 27$

a. $(2x + 3\sqrt{3}i)(2x - 3\sqrt{3}i)$

b. $(2x + 9i)(2x - 3i)$

c. $(2x + 3\sqrt{3})^2$

d. $(2x + 3\sqrt{3})(2x - 3\sqrt{3})$

Explanation:

Step1: Recall the difference of squares formula

The difference of squares formula is \(a^2 - b^2=(a + b)(a - b)\). Also, recall that \(i^2=- 1\), so we can rewrite expressions involving negative square roots using the imaginary unit \(i\).

Step2: Rewrite the given polynomial

We have the polynomial \(4x^{2}+27\). We can rewrite it as \(4x^{2}-(- 27)\) because adding a number is the same as subtracting its negative. Now, \(4x^{2}=(2x)^{2}\) and \(-27 =-(3\sqrt{3})^{2}\), but since we know that \(i^{2}=-1\), we can write \(-27=(3\sqrt{3}i)^{2}\) (because \((3\sqrt{3}i)^{2}=9\times3\times i^{2}=27\times(- 1)=-27\)). So the polynomial \(4x^{2}+27\) can be written as \((2x)^{2}-(3\sqrt{3}i)^{2}\).

Step3: Apply the difference of squares formula

Using the difference of squares formula \(a^{2}-b^{2}=(a + b)(a - b)\) where \(a = 2x\) and \(b=3\sqrt{3}i\), we get \((2x + 3\sqrt{3}i)(2x-3\sqrt{3}i)\).

Let's check option D: \((2x + 3\sqrt{3})(2x - 3\sqrt{3})=(2x)^{2}-(3\sqrt{3})^{2}=4x^{2}-27\), which is not equal to \(4x^{2}+27\). Option C: \((2x + 3\sqrt{3})^{2}=(2x)^{2}+2\times2x\times3\sqrt{3}+(3\sqrt{3})^{2}=4x^{2}+12\sqrt{3}x + 27\), which is not equal to \(4x^{2}+27\). Option B: \((2x + 9i)(2x-3i)=4x^{2}-6xi+18xi - 27i^{2}=4x^{2}+12xi + 27\) (since \(i^{2}=-1\)), which is not equal to \(4x^{2}+27\).

Answer:

A. \((2x + 3\sqrt{3}i)(2x - 3\sqrt{3}i)\)