diff --git a/quarto/precalc/rational_functions.qmd b/quarto/precalc/rational_functions.qmd
index e8ceca8..92aacfc 100644
--- a/quarto/precalc/rational_functions.qmd
+++ b/quarto/precalc/rational_functions.qmd
@@ -239,7 +239,7 @@ bottom = x-1
 quo, rem = divrem(top, bottom)
 ```
 
-The graph of has nothing in common with the graph of the quotient for small $x$
+The graph has nothing in common with the graph of the quotient for small $x$
 
 
 ```{julia}
@@ -265,7 +265,7 @@ $$
 \frac{(x-2)^3\cdot(x-4)\cdot(x-3)}{(x-5)^4 \cdot (x-6)^2}.
 $$
 
-By looking at the powers we can see that the leading term of the numerator will the $x^5$ and the leading term of the denominator $x^6$. The ratio is $1/x^1$. As such, we expect the $y$-axis as a horizontal asymptote:
+By looking at the powers we can see that the leading term of the numerator will the $x^5$ and the leading term of the denominator $x^6$. The ratio is $1/x^1$. As such, we expect the $x$-axis as a horizontal asymptote:
 
 
 #### Partial fractions
@@ -378,7 +378,7 @@ plot(𝒇, -5, 5, ylims=(-20, 20))
 This isn't ideal, as the large values are still computed, just the viewing window is clipped. This leaves the vertical asymptotes still effecting the graph.
 
 
-There is another way, we could ask `Julia` to not plot $y$ values that get too large. This is not a big request.  If instead of the value of `f(x)` - when it is large - -we use `NaN` instead, then the connect-the-dots algorithm will skip those values.
+There is another way, we could ask `Julia` to not plot $y$ values that get too large. This is not a big request.  If instead of the value of `f(x)` - when it is large - we use `NaN` instead, then the connect-the-dots algorithm will skip those values.
 
 
 This was discussed in an earlier section where the `rangeclamp` function was introduced to replace large values of `f(x)` (in absolute values) with `NaN`.
@@ -621,7 +621,7 @@ a, b = 4, 6
 pq = 𝐩 // one(𝐩)
 x = variable(pq)
 d = Polynomials.degree(𝐩)
-numerator(lowest_terms( (x + 1)^2 * pq((a*x + b)/(x + 1))))
+numerator(lowest_terms( (x + 1)^d * pq((a*x + b)/(x + 1))))
 ```
 
 ---
@@ -630,7 +630,7 @@ numerator(lowest_terms( (x + 1)^2 * pq((a*x + b)/(x + 1))))
 Now, why is this of any interest?
 
 
-Mobius transforms are used to map regions into other regions. In this special case, the transform $\phi(x) = (ax + b)/(x + 1)$ takes the interval $[0,\infty]$ and sends it to $[a,b]$  ($0$ goes to $(a\cdot 0 + b)/(0+1) = b$, whereas $\infty$ goes to $ax/x \rightarrow a$). Using this, if $p(u) = 0$, with $q(x) = (x-1)^d p(\phi(x))$,  then setting $u = \phi(x)$ we have $q(x) = (\phi^{-1}(u)+1)^d p(\phi(\phi^{-1}(u))) =  (\phi^{-1}(u)+1)^d \cdot  p(u) =  (\phi^{-1}(u)+1)^d \cdot 0 = 0$. That is, a zero of $p$ in $[a,b]$ will appear as a zero of $q$ in $[0,\infty)$ at $\phi^{-1}(u)$.
+Mobius transforms are used to map regions into other regions. In this special case, the transform $\phi(x) = (ax + b)/(x + 1)$ takes the interval $[0,\infty]$ and sends it to $[a,b]$  ($0$ goes to $(a\cdot 0 + b)/(0+1) = b$, whereas $\infty$ goes to $ax/x \rightarrow a$). Using this, if $p(u) = 0$, with $q(x) = (x+1)^d p(\phi(x))$,  then setting $u = \phi(x)$ we have $q(x) = (\phi^{-1}(u)+1)^d p(\phi(\phi^{-1}(u))) =  (\phi^{-1}(u)+1)^d \cdot  p(u) =  (\phi^{-1}(u)+1)^d \cdot 0 = 0$. That is, a zero of $p$ in $[a,b]$ will appear as a zero of $q$ in $[0,\infty)$ at $\phi^{-1}(u)$.
 
 
 The Descartes rule of signs applied to $q$ then will give a bound on the number of possible roots of $p$ in the interval $[a,b]$. In the example we did, the Mobius transform for $a=4, b=6$ is $15 - x - 11x^2 - 3x^3$ with $1$ sign change, so there must be exactly $1$ real root of $p=(x-1)(x-3)(x-5)$ in the interval $[4,6]$, as we can observe from the factored form of $p$.
@@ -640,7 +640,8 @@ Similarly, we can see there are $2$ or $0$ roots for $p$ in the interval $[2,6]$
 
 
 ```{julia}
-mobius_transformation(𝐩, 2,6)
+p = fromroots([1,3,5])               #  (x-1)⋅(x-3)⋅(x-5) = -15 + 23*x - 9*x^2 + x^3
+mobius_transformation(p, 2,6)
 ```
 
 This observation, along with a detailed analysis provided by [Kobel, Rouillier, and Sagraloff](https://dl.acm.org/doi/10.1145/2930889.2930937) provides a means to find intervals that enclose the real roots of a polynomial.
@@ -662,7 +663,6 @@ Applying these steps to $p$ with an initial interval, say $[0,9]$, we would have
 
 ```{julia}
 #| hold: true
-p = fromroots([1,3,5])               #  (x-1)⋅(x-3)⋅(x-5) = -15 + 23*x - 9*x^2 + x^3
 mobius_transformation(p, 0, 9) # 3
 mobius_transformation(p, 0, 9//2)    # 2
 mobius_transformation(p, 9//2, 9)    # 1 (and done)