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Section 1.7 : Complex Numbers
5. Perform the indicated operation and write your answer in standard form.
\[\left( { - 3 - 9i} \right)\left( {1 + 10i} \right)\]
Hint : You know how to do the operation with polynomials so you can do the operation here! Just recall that you need to be careful to deal with any \(i^{2}\) that might happen to show up in the process.
We know how to multiply two polynomials and so we also know how to multiply two complex numbers. All we need to do is “foil” the two complex numbers to get,
\[\left( { - 3 - 9i} \right)\left( {1 + 10i} \right) = - 3 - 30i - 9i - 90{i^2}\]All we need to do to finish the problem is to recall that \({i^2} = - 1\). Upon using this fact we can finish the problem.
\[\left( { - 3 - 9i} \right)\left( {1 + 10i} \right) = - 3 - 30i - 9i - 90\left( { - 1} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{87 - 39i}}\]