Polynomials

Takes the form $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x + a_{0}, a \in R$ Real Numbers
Actually, this is wrong and $a \in C$ Complex Numbers
Denote the entire set of polynomials over $R$ or $C$ with $R [x]$ or $C [x]$
Degree = number of exponent
- Zero polynomial has undefined degree
Leading coefficient - coefficient with the highest term

Properties

For a degree of n

Turning points
- Max: n
- Min: 0
Even:
- Zeros
  - Max: n
  - Min: 0
- Turning points
  - - leading coefficient
  - as $x \to - \infty, y \to + \infty$
  - as $x \to + \infty, y \to + \infty$
- - leading coefficient
    - as $x \to - \infty, y \to - \infty$
    - as $x \to + \infty, y \to - \infty$
Odd:
- Zeros
  - Max: n
  - Min: 1
- Turning points
  - - leading coefficient
  - as $x \to - \infty, y \to - \infty$
  - as $x \to + \infty, y \to + \infty$
- - leading coefficient
    - as $x \to - \infty, y \to - \infty$
    - as $x \to + \infty, y \to + \infty$

Solving Polynomials

If $f (z)$ has Real Numbers coefficients, then its roots come in conjugate pairs (if $z_{0}$ is a root, so is $\overset{―}{z_{0}}$ )
- Note that this is untrue for non-real coefficients
- Proof: $$
  \begin{align}
  a_{0}+a_{1}z_{0}+\dots+a_{n}z_{0}^{n} & =0 \
  \overline{a_{0}+a_{1}z_{0}+\dots+a_{n}z_{0}^{n}} & =\overline{0} \
  \overline{a_{0}}+\overline{a_{1}}\overline{z_{0}}+\dots+\overline{a_{n}}\overline{z_{0}^{n}} & =0 \
  a_{0}+a_{1}\overline{z_{0}}+\dots+a_{n}\overline{z_{0}}^{n} & =0
  \end

## Family of Functions - Polynomials that have the same roots but different leading coefficient