#import "../handout.typ": *
#import "../macros.typ": *
#import "@preview/cetz:0.3.1"

= Tropical Cubic Polynomials

#problem()
Consider the polynomial $f(x) = x^3 #tp x^2 #tp 3x #tp 6$. \
- sketch a graph of this polynomial
- use this graph to find the roots of $f$
- write (and expand) a product of linear factors with the same graph as $f$.

#graphgrid(none)


#v(1fr)
#pagebreak() // MARK: page

#problem()
Consider the polynomial $f(x) = x^3 #tp x^2 #tp 6x #tp 6$. \
- sketch a graph of this polynomial
- use this graph to find the roots of $f$
- write (and expand) a product of linear factors with the same graph as $f$.

#graphgrid(none)


#v(1fr)

#problem()
Consider the polynomial $f(x) = x^3 #tp 6x^2 #tp 6x #tp 6$. \
- sketch a graph of this polynomial
- use this graph to find the roots of $f$
- write (and expand) a product of linear factors with the same graph as $f$.

#graphgrid(none)


#v(1fr)
#pagebreak() // MARK: page


#problem()
If $f(x) = a x^3 #tp b x^2 #tp c x #tp d$, then $accent(f, macron)(x) = a x^3 #tp B x^2 #tp C x #tp d$ for some $B$ and $C$. \
Using the last three problems, find formulas for $B$ and $C$ in terms of $a$, $b$, $c$, and $d$.

#v(1fr)
#pagebreak() // MARK: page

#problem()
What are the roots of the following polynomial?

#align(
  center,
  box(
    inset: 3mm,
    $
      3 x^6 #tp 4 x^5 #tp 2 x^4 #tp x^3 #tp x^2 #tp 4 x #tp 5
    $,
  ),
)



#v(1fr)
#pagebreak() // MARK: page

#problem()
If
$
  f(x) = c_0 #tp c_1 x #tp c_2 x^2 #tp ... #tp c_n x^n
$
then
$
  accent(f, macron)(x) = c_0 #tp C_1 x #tp C_2 x^2 #tp ... #tp C_(n-1) x^(n-1) #tp c_n x^n
$

#v(2mm)

Find a formula for each $C_i$ in terms of $c_0, c_1, ..., c_n$. \
Then, find formulas for the roots $r_1, r_2, ..., r_n$.

#v(1fr)

#problem()
Can you find a geometric interpretation of these formulas \
in terms of the points $(-i, c_i)$ for $0 <= i <= n$?

#v(0.5fr)