Polynomial

Polynomial Functions
Polynomial functions can be written in the following form: $$f(x) = \sum_{i = 0}^{n} a_{i} x^{i}.$$

Roots
We can express a quadratic polynomial function as $$p(x) = ax^2+bx+c$$ According to the Fundamental Theorem of Algebra, a quadratic polynomial function has two roots, i.e. two numbers $$r_1$$ and $$r_2$$ such that $$p(r_1) = p(r_2) = 0$$. It can be shown that $${r_1,r_2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

The sum and product of the roots can be expressed using the coefficients. In the following, D denotes $$\sqrt{b^2-4ac}$$.

$$r_1 + r_2 = \frac{-b-D}{2a} + \frac{-b+D}{2a}$$ $$= \frac{-2b-D+D}{2a} = -\frac{b}{a}$$

$$r_1r_2 = \frac{-b-D}{2a}\cdot\frac{-b+D}{2a}$$ $$= \frac{b^2-bD+bD-D^2}{4a^2} = \frac{b^2-D^2}{4a^2}$$ $$= \frac{b^2-b^2-4ac}{4a^2} = \frac{4ac}{4a^2} = \frac{c}{a}$$

Factorization
Any polynomial can be factorized $$\sum_{i = 0}^{n} a_{i} x^{i} = \Pi_{}^{}$$

For example, a quadratic polynomial can be factorized:

$$ax^2+bx+c = (x-r_1)(x-r_2)$$


 * S.O.S. Math: Factoring and Roots of Polynomials explains the very simple connection between factorization and roots.