Added section 3 problems

Advanced/Tropical Polynomials/main.typ

@@ -17,3 +17,6 @@
 #pagebreak()
 
 #include "parts/01 polynomials.typ"
+#pagebreak()
+
+#include "parts/02 cubic.typ"

Advanced/Tropical Polynomials/parts/02 cubic.typ

@@ -0,0 +1,89 @@
+#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)