## Mathematical Background Outline

- Polynomials
- Terminology: coefficients, terms, factors, degree
- Relating roots and coefficients of quadratic polynomials
- Average of roots and average of roots of derivative are the same

- Complex Numbers
- Basic definitions and properties
- Polar form and geometric interpretations: writing any complex number as
*x + iy = re*^{iq}; multiplying and adding complex numbers geometrically
- Geometric condition for
*zw* to be a negative real, where *z* and *w* are complex numbers

- Linear and geometric transformations of the plane
- Translation, rotation, and scaling figures
- Representing transformations with matrices
- Shears and one directional scaling
- Properties of linear transformations

- Ellipses
- Overview concepts of ellipses and conic sections
- Definition of ellipse as a deformed circle
- Standard equation for an ellipse:
- Equivalent definition: set of points the sum of whose distances from two given points is fixed
- Linear transformations take ellipses to ellipses
- First uniqueness property: Given two points
*E* and *F* and a point *P* not on the segment between *E* and *F*, there is a unique ellipse through *P* having foci at *E* and *F*
- Second uniqueness property: Given two points
*E* and *F* and a line *L* not intersecting the segment between *E* and *F*, there is a unique ellipse tangent to *L* having foci at *E* and *F*
- Optical Property: At any point
*P* of an ellipse, the lines to *P* from the foci make equal acute angles with the tangent to the ellipse at *P*.
- Extended Optical Property: At any point
*P* outside of an ellipse, the lines to *P* from the foci make equal acute angles with the two tangent lines to the ellipse from the point *P*.

- Every triangle has a unique inscribed ellipse, tangent at midpoints