Difference of Squares gives which complex factor expression for [tex]\( x^2 + 11 \)[/tex]?

A. [tex]\((x + i\sqrt{11})(x + i\sqrt{11})\)[/tex]
B. [tex]\((x + i\sqrt{11})(x - i\sqrt{11})\)[/tex]
C. [tex]\((x + i\sqrt[4]{11})(x - i\sqrt[4]{11})\)[/tex]
D. [tex]\((x + i\sqrt{11})(x - i\sqrt[4]{11})^2\)[/tex]



Answer :

To factor the given expression [tex]\( x^2 + 11 \)[/tex] into its complex factors, we use the concept of the difference of squares and the fact that [tex]\( -1 \)[/tex] can introduce complex units.

Recall that:

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

where [tex]\( i \)[/tex] is the imaginary unit and [tex]\( a \)[/tex] is a positive real number. In this case, [tex]\( a^2 = 11 \)[/tex], so [tex]\( a = \sqrt{11} \)[/tex].

Thus, we can express [tex]\( x^2 + 11 \)[/tex] as:

[tex]\[ x^2 + (\sqrt{11})^2 \][/tex]

Using the above factorization formula, we get:

[tex]\[ x^2 + 11 = (x + \sqrt{11}i)(x - \sqrt{11}i) \][/tex]

Therefore, the correct factorization of [tex]\( x^2 + 11 \)[/tex] into its complex factors is:

[tex]\[ (x + \sqrt{11}i)(x - \sqrt{11}i) \][/tex]

So, the correct answer is:

D. [tex]\( (x + i\sqrt{11})(x - i\sqrt{11}) \)[/tex]