Quadratic Polynomials

A monic quadratic polynomial has the form p(x) = x2+bx+c. That was in HTML. In MathML that equation looks like px = x2 + bx+c. The roots of this polynomial have the form r= -b 2 ± b2- 4c 2 . If we suppose that b and c are in ℝ, then the roots are both real, or are complex conjugates. In any case, the sum of the roots is i=1 2 ri. Now, every polynomial has a so-called companion matrix associated with it. In this case, one form of the companion matrix is Cp= 0 -c 1 -b . The reason this is called the companion matrix is that det xI- Cp = px. It follows that the roots of px are exactly the eigenvalues of Cp.

Here is some SVG that has nothing to do with companion matrices.

Vertex (a,0)