### Video Transcript

Factor 𝑥 squared plus nine over
the complex numbers.

In order to answer this question,
we need to recall the rule we use when trying to factor the sum of two squares. We can do this using complex
numbers such that 𝑎 squared plus 𝑏 squared is equal to 𝑎 plus 𝑏𝑖 multiplied by
𝑎 minus 𝑏𝑖. We can prove this rule by
distributing our parentheses or expanding the brackets using the FOIL method.

Multiplying the first terms in our
parentheses gives us 𝑎 squared. Multiplying the outer terms gives
us negative 𝑎𝑏𝑖. Multiplying the inner terms, we
have positive 𝑎𝑏𝑖. And finally multiplying the last
terms, we have negative 𝑏 squared 𝑖 squared. We recall from our knowledge of
complex numbers that 𝑖 squared is equal to negative one. As the 𝑎𝑏𝑖 terms cancel, we are
left with 𝑎 squared minus 𝑏 squared multiplied by negative one. This simplifies to 𝑎 squared plus
𝑏 squared.

Returning to our question, we see
that our value of 𝑎 is 𝑥 and our value of 𝑏 is three as three squared is equal to
nine. 𝑥 squared plus nine can,
therefore, be factored over the complex numbers to give us 𝑥 plus three 𝑖
multiplied by 𝑥 minus three 𝑖.