em dash; sentence case

quarto/precalc/polynomial_roots.qmd

@@ -448,12 +448,12 @@ To get the numeric approximation, we can broadcast:
 N.(solveset(p ~ 0, x))
 ```
 
-(There is no need to call `collect` -- though you can -- as broadcasting over a set falls back to broadcasting over the iteration of the set and in this case returns a vector.)
+(There is no need to call `collect`---though you can---as broadcasting over a set falls back to broadcasting over the iteration of the set and in this case returns a vector.)
 
 ## Do numeric methods matter when you can just graph?
 
 
-It may seem that certain practices related to roots of polynomials are unnecessary as we could just graph the equation and look for the roots. This feeling is perhaps motivated by the examples given in textbooks to be worked by hand, which necessarily focus on smallish solutions. But, in general, without some sense of where the roots are, an informative graph itself can be hard to produce. That is, technology doesn't displace thinking--it only supplements it.
+It may seem that certain practices related to roots of polynomials are unnecessary as we could just graph the equation and look for the roots. This feeling is perhaps motivated by the examples given in textbooks to be worked by hand, which necessarily focus on smallish solutions. But, in general, without some sense of where the roots are, an informative graph itself can be hard to produce. That is, technology doesn't displace thinking---it only supplements it.
 
 
 For another example, consider the polynomial $(x-20)^5 - (x-20) + 1$. In this form we might think the roots are near $20$. However, were we presented with this polynomial in expanded form: $x^5 - 100x^4 + 4000x^3 - 80000x^2 + 799999x - 3199979$, we might be tempted to just graph it to find roots.  A naive graph might be to plot over $[-10, 10]$: