TITLE: Why is finding the roots of a polynomial equation so important? What is to gain?
QUESTION [11 upvotes]: I have just started a pre calculus class, and our first lessons have been reviews on polynomial equation, quadratics and finding roots or solutions to equations. The topic is fairly simple but I just have to know why is it so important to find an X value of a function that results in it y value to become zero? Why do mathematicians go to such great lengths like creating a quadratic formula? There must be some reason and my lessons have failed to tell why.

REPLY [5 votes]: In a word:  factoring.  In logical terms:  $ab=0\implies a=0$ or $b=0$.
To the extent that algebra concerns itself with setting up and solving polynomial equations, bringing everything to one side, leaving $0$ on the other side, is often an effective way to find all the solutions of an equation -- and, importantly, to know you haven't overlooked any.  For example, consider the equation
$$(x^2+11)x=6(x^2+1)$$
If you play around with it, you might notice that $x=1$ is a solution, since $12\cdot1=6\cdot2$.  If you play some more, you might find a couple of other solutions.  And if you know some theory, you might know when you've found all the solutions.  But it becomes systematic, and easier, if you rewrite the equation as
$$x^3-6x^2+11x-6=0$$
The observation that $x=1$ is a solution means that the cubic on the left hand side factors:
$$(x-1)(x^2-5x+6)=0$$
In fact, the cubic in this case factors completely, giving us
$$(x-1)(x-2)(x-3)=0$$
so this tells us that $\{1,2,3\}$ is the complete solution set to the equation $(x^2+11)x=6(x^2+1)$.
Not every equation works out so nicely, of course.  But at the very least, setting things up to solve for roots is a systematic way of chipping away at problems involving polynomials.