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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> 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).

I would actually assert that Algebra is altogether broadly described as a set of rules for manipulating expressions without altering the equality.

The fact that this manifests as an ability to convert 2x = 10 to x = 5 is only of importance when we assign physical representation to these equations otherwise there is no inherent significance that x = 5 has over 2x = 10.

The factoring of polynomials has more applications in advanced sciences and computing. Linear algebra and differential equations see extensive factoring. Balancing chemical equations? Factoring polynomials. Does every problem require use of the quadratic equation, absolutely not. Take a calculus class and you’ll thank your teachers for the ability to rewrite data.