If the practical use of algebra and math broadly is to calculate an unknown answer then why is factoring polynomials still taught if it is merely a means to rewrite data which one already has?

Question

If the practical use of algebra and math broadly is to calculate an unknown answer then why is factoring polynomials still taught if it is merely a means to rewrite data which one already has?

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Answer 1

> why is factoring polynomials still taught if it is merely a means to rewrite data which one already has?

Well, it is *not* merely a means to rewrite data one already has!
The roots of a polynomial give important information that are not evident just from the original form of the polynomial. Here are a few examples.

1) Knowing whether all the roots of certain polynomial are negative (or more broadly have negative real part) comes up in stability theory of dynamical systems.

2) Explicit formulas for sequences satisfying a linear recursion (like an explicit formula for the Fibonacci numbers) need roots of polynomials.

3) Factoring polynomials (really, finding roots) is a step in error correction for certain codes (BCH codes).