Question Video: Factoring the Sum of Two Squares Mathematics

Factor 𝑥² + 9 over the complex numbers.

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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.

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