FIR draft

src/Advanced/Tropical Polynomials/parts/00 arithmetic.typ

@@ -62,8 +62,7 @@ Let's expand $#sym.RR$ to include a tropical additive identity.
 #problem()
 Do tropical additive inverses exist? \
 #note([
-  Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$? \
-  Remember that $#sym.infinity$ is the additive identity.
+  Is there an inverse $y$ for every $x$ so that $x #tp y = #sym.infinity$?
 ])
 
 #solution([
@@ -278,7 +277,7 @@ Fill the following tropical addition and multiplication tables
 
 #problem()
 Expand and simplify $f(x) = (x #tp 2)(x #tp 3)$, then evaluate $f(1)$ and $f(4)$ \
-#hint([Adjacent parenthesis imply tropical multiplication])
+Adjacent parenthesis imply tropical multiplication
 
 #solution([
   $

src/Advanced/Tropical Polynomials/parts/01 polynomials.typ

@@ -20,7 +20,7 @@ for some nonnegative integer $n$ and coefficients $c_0, c_1, ..., c_n$. \
 The _degree_ of a polynomial is the largest $n$ for which $c_n$ is nonzero.
 
 #theorem()
-The _fundamental theorem of algebra_ states that any non-constant polynomial with real coefficients
+The _fundamental theorem of algebra_ implies that any non-constant polynomial with real coefficients
 can be written as a product of polynomials of degree 1 or 2 with real coefficients.
 
 #v(2mm)
@@ -30,8 +30,8 @@ can be written as $(x^2 + 2x+5)(x-2)(x+4)(x+4)$
 
 #v(2mm)
 A similar theorem exists for polynomials with complex coefficients. \
-These coefficients may be found using the _roots_ of this polynomial. \
-As you already know, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on appropriate roots found by computers.
+These coefficients may be found using the roots of this polynomial. \
+As it turns out, there are formulas that determine the roots of quadratic, cubic, and quartic #note([(degree 2, 3, and 4)]) polynomials. There are no formulas for the roots of polynomials with larger degrees---in this case, we usually rely on approximate roots found by computers.
 
 #v(2mm)
 In this section, we will analyze tropical polynomials:
@@ -100,7 +100,7 @@ In other words, find $r$ and $s$ so that
   ),
 )
 
-#note([Naturally, we will call $r$ and $s$ the _roots_ of $f$.])
+we will call $r$ and $s$ the _roots_ of $f$.
 
 #solution([
   Because $(x #tp r)(x #tp s) = x^2 #tp (r #tp s)x #tp s r$, we must have $r #tp s = 1$ and $r #tm s = 4$. \
@@ -116,8 +116,7 @@ In other words, find $r$ and $s$ so that
 #v(1fr)
 
 #problem()
-Can you see the roots of this polynomial in the graph? \
-#hint([Yes, you can. What "features" do the roots correspond to?])
+How can we use the graph to determine these roots?
 
 #solution([The roots are the corners of the graph.])
 
@@ -317,7 +316,7 @@ into linear factors.
 #v(2mm)
 
 Whenever we say "the roots of $f$", we really mean "the roots of $accent(f, macron)$." \
-Also, $f$ and $accent(f, macron)$ might be the same polynomial.
+$f$ and $accent(f, macron)$ might be the same polynomial.
 
 #problem()
 If $f(x) = a x^2 #tp b x #tp c$, then $accent(f, macron)(x) = a x^2 #tp B x #tp c$ for some $B$. \