make pdf file generation work

quarto/precalc/polynomials_package.qmd

@@ -249,13 +249,13 @@ A line, $y=mx+b$ can be a linear polynomial or a constant depending on $m$, so w
 Knowing we can succeed, we approach the problem of $3$ points, say $(x_0, y_0)$, $(x_1,y_1)$, and $(x_2, y_2)$. There is a polynomial $p = a\cdot x^2 + b\cdot x + c$ with $p(x_i) = y_i$. This gives $3$ equations for the $3$ unknown values $a$, $b$, and $c$:
 
 
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+
 \begin{align*}
 a\cdot x_0^2 + b\cdot x_0 + c &= y_0\\
 a\cdot x_1^2 + b\cdot x_1 + c &= y_1\\
 a\cdot x_2^2 + b\cdot x_2 + c &= y_2\\
 \end{align*}
-$$
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 Solving this with `SymPy` is tractable. A comprehension is used below to create the $3$ equations; the `zip` function is a simple means to iterate over $2$ or more iterables simultaneously:
 
@@ -388,13 +388,13 @@ radioq(choices, answ, keep_order=true)
 Consider the polynomial $p(x) = a_1 x - a_3 x^3 + a_5 x^5$ where
 
 
-$$
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 \begin{align*}
 a_1 &= 4(\frac{3}{\pi} - \frac{9}{16}) \\
 a_3 &= 2a_1 -\frac{5}{2}\\
 a_5 &= a_1 - \frac{3}{2}.
 \end{align*}
-$$
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   * Form the polynomial `p` by first computing the $a$s and forming `p=Polynomial([0,a1,0,-a3,0,a5])`
   * Form the polynomial `q` by these commands `x=variable(); q=p(2x/pi)`
@@ -544,13 +544,13 @@ This last answer is why $p$ is called an *interpolating* polynomial and this que
 The Chebyshev ($T$) polynomials are polynomials which use a different basis from the standard basis. Denote the basis elements $T_0$, $T_1$, ... where we have $T_0(x) = 1$, $T_1(x) = x$, and for bigger indices $T_{i+1}(x) = 2xT_i(x) - T_{i-1}(x)$. The first others are then:
 
 
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 \begin{align*}
 T_2(x) &= 2xT_1(x) - T_0(x) = 2x^2 - 1\\
 T_3(x) &= 2xT_2(x) - T_1(x) = 2x(2x^2-1) - x = 4x^3 - 3x\\
 T_4(x) &= 2xT_3(x) - T_2(x) = 2x(4x^3-3x) - (2x^2-1) = 8x^4 - 8x^2 + 1
 \end{align*}
-$$
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 With these definitions what is the polynomial associated to the coefficients $[0,1,2,3]$ with this basis?