Select the correct answer.

Which expression is equivalent to this polynomial?

[tex]\[ x^2 + 12 \][/tex]

A. [tex]\((x + 2\sqrt{3}i)(x - 2\sqrt{3}i)\)[/tex]

B. [tex]\((x + 6i)(x - 6i)\)[/tex]

C. [tex]\((x + 2\sqrt{3})^2\)[/tex]

D. [tex]\((x + 2\sqrt{3})(x - 2\sqrt{3})\)[/tex]



Answer :

To determine which expression is equivalent to the polynomial [tex]\(x^2 + 12\)[/tex], let's factor it completely.

We start by using the difference of squares identity: [tex]\(a^2 - b^2 = (a+b)(a-b)\)[/tex]. However, since our polynomial is [tex]\(x^2 + 12\)[/tex], we need to incorporate complex numbers, as the sum of squares can be factored in the form involving complex numbers:

[tex]\[ x^2 + a^2 = (x + ai)(x - ai) \][/tex]

In this case, we can rewrite 12 in the form of [tex]\(a^2\)[/tex] where [tex]\(a = 2\sqrt{3}\)[/tex]. Therefore, [tex]\(12 = (2\sqrt{3})^2\)[/tex]:

Now, our polynomial becomes:

[tex]\[ x^2 + (2\sqrt{3})^2 \][/tex]

Using the complex number factorization, we have:

[tex]\[ x^2 + (2\sqrt{3})^2 = (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) \][/tex]

Thus, the equivalent expression is:

[tex]\[ (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) \][/tex]

Comparing this with the given options, we see that Option A is:

[tex]\[ (x + 2\sqrt{3}i)(x - 2\sqrt{3}i) \][/tex]

Therefore, the correct answer is:

[tex]\[ \boxed{1} \][/tex]