Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Factor a sum of squares over the complex numbers using \(a^2+b^2=(a+bi)(a-bi)\)
Problem
Factor \(x^{2}+9\) over the complex numbers by rewriting it as \(x^{2}-(3i)^{2}\) and applying the difference-of-squares pattern.
Big Picture
What this problem is really about
A sum of real squares becomes a difference of squares once the second square is written with the imaginary unit. We’ll create the negative square, rewrite the expression in difference form, factor into conjugates, and multiply back to confirm that the cross terms cancel and the sign returns to a sum.
Inside Gozunta
Turn the preview into practice.
More than a preview
Gozunta lets you study this problem—and more than 10,000 others.
Create targeted learning sessions, use hints, check your answer, read the walkthrough, watch the video, print the work, and track your progress.
