Polynomial functions can be written in the following form: <math>f(x) = \sum_{i = 0}^{n} a_{i} x^{i}.</math>
We can express a quadratic polynomial function as <math>p(x) = ax^2+bx+c</math> According to the Fundamental Theorem of Algebra, a quadratic polynomial function has two roots, i.e. two numbers <math>r_1</math> and <math>r_2</math> such that <math>p(r_1) = p(r_2) = 0</math>. It can be shown that <math>{r_1,r_2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>
The sum and product of the roots can be expressed using the coefficients. In the following, D denotes <math>\sqrt{b^2-4ac}</math>.
<math>r_1 + r_2 = \frac{-b-D}{2a} + \frac{-b+D}{2a}</math> <math>= \frac{-2b-D+D}{2a} = -\frac{b}{a}</math>
<math>r_1r_2 = \frac{-b-D}{2a}\cdot\frac{-b+D}{2a}</math> <math>= \frac{b^2-bD+bD-D^2}{4a^2} = \frac{b^2-D^2}{4a^2}</math> <math>= \frac{b^2-b^2-4ac}{4a^2} = \frac{4ac}{4a^2} = \frac{c}{a}</math>
Any polynomial can be factorized <math>\sum_{i = 0}^{n} a_{i} x^{i} = \Pi_{}^{}</math>
For example, a quadratic polynomial can be factorized:
<math>ax^2+bx+c = (x-r_1)(x-r_2)</math>