Why do we study patterns in mathematics?

A big part of math is pattern recognition. A lot of homework is just fighting with stuff so you will remember the pattern when you see it again. The most common pattern is (x + a) * (x + b) = x^2 + (a + b)x + ab and the special case (x + a) * (x - a) = x^2 - a^2. When you spot the pattern you can just write the answer from memory.

Now study this page until it seems obvious: https://en.wikipedia.org/wiki/Polynomial_remainder_theorem

Example: You run across something of the form y = x^2 + a^2 and you notice the vague resemblance to the special case above. Then you recall that i^2 = -1 so (-i)^2 = 1. then you write x^2 + a^2 = (x + ai) * (x - ai) just because you recognized the pattern.