Polynomial Roots
Use roots, factors and coefficients to analyse polynomials.
If \(x=\alpha\) is a root of \(P(x)\), then \((x-\alpha)\) is a factor of \(P(x)\).
Factor theorem\[P(\alpha)=0\quad\Longleftrightarrow\quad (x-\alpha)\text{ is a factor of }P(x)\]
Quadratic root relationships
For \(ax^2+bx+c=0\), if the roots are \(\alpha\) and \(\beta\), then
\[\alpha+\beta=-\frac ba\qquad \alpha\beta=\frac ca\]
HL extension
Similar relationships exist for cubic and higher-degree polynomials. These are useful for constructing equations from known roots.