Disclaimer
For the algebra section only, I assume everyone reading this guide is algebraically sound. This section thus contains only little tips and hacks I have found useful.
Polynomials
Root finding
- Finding the root of a 1st-order polynomial is trivial
- Finding the root of a 2nd-order polynomial uses the quadratic equation.
\(\text{for a polynomial of form: } ax^2 + bx + c = 0\)
\(\text{use: } x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
derived from completing the square. to get the ball rolling, divide both sides by $a$, move $c/a$, add $b^2/4a^2$ to both sides…
- Finding the root of a 3rd and 4th order polynomial require advanced functions that are not practically seen, however, the solutions remain analytical.
- Find the root of a degree 5+ polynomial is no longer analytical. Numerical solvers must be used.
- The matlab $\texttt{root(p)}$ function or python $\texttt{np.roots(coefficients)}$ functions are your best friend for degree 3+ polynomials