建立二次方程因式分解工作表：如何摆脱不需要的零和一？

Question 1

要选择在范围内\b且在范围内的\d随机不同数字，我将使用\d1..10\d-10,...,-1

 \pgfmathsetmacro{\d}{int(random(1,10)}
 \pgfmathsetmacro{\b}{int(random(1,9))}

现在\b用-\bif\b<\d和-1-\belse 替换。

对于系数的打印问题，您只需要另一个\ifnum分支。事实上，您需要两个分支，因为-1x您不想打印，而是想打印-x。

这是经过这些修改的 MWE：

\documentclass{article}

\usepackage[margin=1.2cm]{geometry}
\usepackage{ifthen}
\usepackage{multicol}
    \setlength{\columnsep}{1.5cm}
\usepackage{pgf}
\usepackage{pgffor}

\setlength{\parindent}{0pt}

\pagestyle{empty}

\pgfmathsetseed{\number\pdfrandomseed}


% How to make sure that we don't get a difference of squares, i.e. if \b=-\c, then the middle term will be zero and it's a DOS.
\newcommand{\InitVariables}
{%
 \pgfmathsetmacro{\d}{int(random(1,10)}
 \pgfmathsetmacro{\b}{int(random(1,9))}
 \ifnum\b<\d\pgfmathsetmacro{\b}{int(-\b)}%
 \else\pgfmathsetmacro{\b}{int(-1-\b)}%
 \fi
 \pgfmathsetmacro{\B}{int(\d+\b)}
 \pgfmathsetmacro{\C}{int(\b*\d)}
}

\newcommand{\trinomial}
{%
\InitVariables
\((x\b)(x+\d)=x^2 \ifnum\B<0\ifnum\B=-1-\else\B\fi\else+\ifnum\B>1\B\fi\fi x\C\)
}

\newcommand{\ManyTrinomials}[1]
{%
\foreach \x in {1,2,3,...,#1}
{\trinomial \\}
}

\begin{document}

\ManyTrinomials{50}

\end{document}

以及输出的随机快照：

根据您正在做的其他事情，编写一个通用宏来漂亮地打印多项式可能会更好。

编辑

这是一个带有\printpolynomial“漂亮打印多项式”宏的版本。该宏采用逗号分隔的系数列表，因此

\printpolynomial{3,-2,1}% --> 3x^2-2x+1

请注意，该宏假定它处于数学模式，因此上述内容确实应该是 $\printpolynomial{3,-2,1}$ 。

为此，代码首先对系数进行一次试运行，以求得多项式的最大次数，称为\polydeg——实际上，这是次数加一。然后，它再次遍历系数并打印它们。使用宏\plusseperator来处理许多首项系数为 0 的烦人情况：

\printpolynomial{0,0,0,0,0,2,-2}%. -> 2x-2

在极端情况下，\printpolynomial{0,0,0,0,0}会打印0。除了担心系数1和之外-1，在处理多项式（如）时，指数和还会产生其他复杂情况\printpolynomial{1,1}，其中常数项的系数确实需要打印。

作为最后的奖励，该宏接受一个可选参数，即行列式的名称。所以

\printpolynomial[y]{3,-4,2,1}%. --> 3y^3-4y^2+2y+1

如果打印指数增加的多项式，则可以稍微缩短宏；即以的形式a_0+a_1x+a_2x^2+...+a_nx^n。

修改后的代码如下：

\documentclass{article}

\usepackage[margin=1.2cm]{geometry}
\usepackage{ifthen}
\usepackage{multicol}
    \setlength{\columnsep}{1.5cm}
\usepackage{pgf}
\usepackage{pgffor}

\setlength{\parindent}{0pt}

\pagestyle{empty}

\pgfmathsetseed{\number\pdfrandomseed}

% usage: \printpolnomial[name of indeterminate]{a_n, ... , a_1, a_0}
\newcommand\printpolynomial[2][x]{%
  % need to loop over #2 once to find the degree (actually, polydeg=degree+1)
  \foreach \coeff [count=\polydeg, remember=\polydeg] in {#2} {}
  \let\plusseperator\relax%  as the leading coefficients may be zero use this instead of +
  \foreach \coeff [count=\term, evaluate=\term as \pdeg using int(\polydeg-\term)] in {#2} {%
     \ifnum\coeff=0\relax% do nothing
     \else
       \ifnum\coeff>0\relax%
          \plusseperator% print a + if some terms have already appeared
          \ifnum\coeff=1\relax%
            \ifnum\pdeg=0\relax1\fi% print 1 if constant term
          \else\coeff% print coefficient if it is greater than 1
          \fi%
       \else\ifnum\coeff<0%
          \ifnum\coeff=-1\relax%
             \ifnum\pdeg=0\relax-1\else-\fi% print -1 if constant term
           \else\coeff% print \coeff
           \fi%
         \fi%
       \fi%
       \ifnum\pdeg=0\relax%
       \else%
         \ifnum\pdeg=1\relax #1\else {#1}^{\pdeg}\fi% print the power of #1 = x
       \fi%
       \gdef\plusseperator{+}% change \plusseperator to + as polynomial is non-zero
     \fi%
   }%
   \ifx\plusseperator\relax 0\fi% zero polynomial
}


% How to make sure that we don't get a difference of squares, i.e.
% if \b=-\d, then the middle term will be zero and it's a DOS.
\newcommand{\InitVariables}
{%
 \pgfmathsetmacro{\d}{int(random(1,10)}
 \pgfmathsetmacro{\b}{int(random(1,9))}
 \ifnum\b<\d\pgfmathsetmacro{\b}{int(-\b)}%
 \else\pgfmathsetmacro{\b}{int(-1-\b)}%
 \fi
 \pgfmathsetmacro{\B}{int(\d+\b)}
 \pgfmathsetmacro{\C}{int(\b*\d)}
}

\newcommand{\trinomial}
{%
\InitVariables
$(\printpolynomial{1,\b})(\printpolynomial{1,\d})
    = \printpolynomial{1,\B,\C}$
}

\newcommand{\ManyTrinomials}[1]
{%
\foreach \x in {1,2,3,...,#1}
    {\trinomial\\}
}

\begin{document}

$\printpolynomial{2,1,3,-2,0}$

$\printpolynomial{0,1,3,-2,0}$

$\printpolynomial{0,0,0,0,0}$

\ManyTrinomials{50}

\end{document}

Answer

要选择在范围内\b且在范围内的\d随机不同数字，我将使用\d1..10\d-10,...,-1

 \pgfmathsetmacro{\d}{int(random(1,10)}
 \pgfmathsetmacro{\b}{int(random(1,9))}

现在\b用-\bif\b<\d和-1-\belse 替换。

对于系数的打印问题，您只需要另一个\ifnum分支。事实上，您需要两个分支，因为-1x您不想打印，而是想打印-x。

这是经过这些修改的 MWE：

\documentclass{article}

\usepackage[margin=1.2cm]{geometry}
\usepackage{ifthen}
\usepackage{multicol}
    \setlength{\columnsep}{1.5cm}
\usepackage{pgf}
\usepackage{pgffor}

\setlength{\parindent}{0pt}

\pagestyle{empty}

\pgfmathsetseed{\number\pdfrandomseed}


% How to make sure that we don't get a difference of squares, i.e. if \b=-\c, then the middle term will be zero and it's a DOS.
\newcommand{\InitVariables}
{%
 \pgfmathsetmacro{\d}{int(random(1,10)}
 \pgfmathsetmacro{\b}{int(random(1,9))}
 \ifnum\b<\d\pgfmathsetmacro{\b}{int(-\b)}%
 \else\pgfmathsetmacro{\b}{int(-1-\b)}%
 \fi
 \pgfmathsetmacro{\B}{int(\d+\b)}
 \pgfmathsetmacro{\C}{int(\b*\d)}
}

\newcommand{\trinomial}
{%
\InitVariables
\((x\b)(x+\d)=x^2 \ifnum\B<0\ifnum\B=-1-\else\B\fi\else+\ifnum\B>1\B\fi\fi x\C\)
}

\newcommand{\ManyTrinomials}[1]
{%
\foreach \x in {1,2,3,...,#1}
{\trinomial \\}
}

\begin{document}

\ManyTrinomials{50}

\end{document}

以及输出的随机快照：

根据您正在做的其他事情，编写一个通用宏来漂亮地打印多项式可能会更好。

编辑

这是一个带有\printpolynomial“漂亮打印多项式”宏的版本。该宏采用逗号分隔的系数列表，因此

\printpolynomial{3,-2,1}% --> 3x^2-2x+1

请注意，该宏假定它处于数学模式，因此上述内容确实应该是 $\printpolynomial{3,-2,1}$ 。

为此，代码首先对系数进行一次试运行，以求得多项式的最大次数，称为\polydeg——实际上，这是次数加一。然后，它再次遍历系数并打印它们。使用宏\plusseperator来处理许多首项系数为 0 的烦人情况：

\printpolynomial{0,0,0,0,0,2,-2}%. -> 2x-2

在极端情况下，\printpolynomial{0,0,0,0,0}会打印0。除了担心系数1和之外-1，在处理多项式（如）时，指数和还会产生其他复杂情况\printpolynomial{1,1}，其中常数项的系数确实需要打印。

作为最后的奖励，该宏接受一个可选参数，即行列式的名称。所以

\printpolynomial[y]{3,-4,2,1}%. --> 3y^3-4y^2+2y+1

如果打印指数增加的多项式，则可以稍微缩短宏；即以的形式a_0+a_1x+a_2x^2+...+a_nx^n。

修改后的代码如下：

\documentclass{article}

\usepackage[margin=1.2cm]{geometry}
\usepackage{ifthen}
\usepackage{multicol}
    \setlength{\columnsep}{1.5cm}
\usepackage{pgf}
\usepackage{pgffor}

\setlength{\parindent}{0pt}

\pagestyle{empty}

\pgfmathsetseed{\number\pdfrandomseed}

% usage: \printpolnomial[name of indeterminate]{a_n, ... , a_1, a_0}
\newcommand\printpolynomial[2][x]{%
  % need to loop over #2 once to find the degree (actually, polydeg=degree+1)
  \foreach \coeff [count=\polydeg, remember=\polydeg] in {#2} {}
  \let\plusseperator\relax%  as the leading coefficients may be zero use this instead of +
  \foreach \coeff [count=\term, evaluate=\term as \pdeg using int(\polydeg-\term)] in {#2} {%
     \ifnum\coeff=0\relax% do nothing
     \else
       \ifnum\coeff>0\relax%
          \plusseperator% print a + if some terms have already appeared
          \ifnum\coeff=1\relax%
            \ifnum\pdeg=0\relax1\fi% print 1 if constant term
          \else\coeff% print coefficient if it is greater than 1
          \fi%
       \else\ifnum\coeff<0%
          \ifnum\coeff=-1\relax%
             \ifnum\pdeg=0\relax-1\else-\fi% print -1 if constant term
           \else\coeff% print \coeff
           \fi%
         \fi%
       \fi%
       \ifnum\pdeg=0\relax%
       \else%
         \ifnum\pdeg=1\relax #1\else {#1}^{\pdeg}\fi% print the power of #1 = x
       \fi%
       \gdef\plusseperator{+}% change \plusseperator to + as polynomial is non-zero
     \fi%
   }%
   \ifx\plusseperator\relax 0\fi% zero polynomial
}


% How to make sure that we don't get a difference of squares, i.e.
% if \b=-\d, then the middle term will be zero and it's a DOS.
\newcommand{\InitVariables}
{%
 \pgfmathsetmacro{\d}{int(random(1,10)}
 \pgfmathsetmacro{\b}{int(random(1,9))}
 \ifnum\b<\d\pgfmathsetmacro{\b}{int(-\b)}%
 \else\pgfmathsetmacro{\b}{int(-1-\b)}%
 \fi
 \pgfmathsetmacro{\B}{int(\d+\b)}
 \pgfmathsetmacro{\C}{int(\b*\d)}
}

\newcommand{\trinomial}
{%
\InitVariables
$(\printpolynomial{1,\b})(\printpolynomial{1,\d})
    = \printpolynomial{1,\B,\C}$
}

\newcommand{\ManyTrinomials}[1]
{%
\foreach \x in {1,2,3,...,#1}
    {\trinomial\\}
}

\begin{document}

$\printpolynomial{2,1,3,-2,0}$

$\printpolynomial{0,1,3,-2,0}$

$\printpolynomial{0,0,0,0,0}$

\ManyTrinomials{50}

\end{document}

Question 2

sagetex如果你愿意使用文档包，你可以使编码变得更容易这里。除了计算机代数系统的计算能力之外，您还可以通过 latex() 命令轻松编写 Python 命令，甚至内置命令来使输出变得漂亮。下面包含 latex(factor(poly)) 的代码行告诉 Sage 对创建的多项式进行因式分解，并以 LaTeX 代码表示结果。

\documentclass{article}%
\usepackage{sagetex}
\begin{document}
\begin{sagesilent}
x = var('x')
Problems = r"Factor each quadratic:\\ \begin{enumerate}"
for i in range(0,20):
    r1 = Integer(randint(-10,10))
    r2 = Integer(randint(-10,10))
    while r1 == -r2:
        r2 = Integer(randint(-10,10))
    poly = expand((x-r1)*(x-r2))
    Problems += r"\item $%s = %s$"%(latex(poly),latex(factor(poly)))
Problems += r"\end{enumerate}"
\end{sagesilent}
\sagestr{Problems}
\end{document}

输出显示在SagemathCloud 中运行：

习惯使用 Sage 和 sagemath 软件包可以让各种数学工作表变得更容易——Sage 可处理所有级别的数学您需要在计算机上本地安装 Sage 或免费SagemathCloud 帐户。

Answer

sagetex如果你愿意使用文档包，你可以使编码变得更容易这里。除了计算机代数系统的计算能力之外，您还可以通过 latex() 命令轻松编写 Python 命令，甚至内置命令来使输出变得漂亮。下面包含 latex(factor(poly)) 的代码行告诉 Sage 对创建的多项式进行因式分解，并以 LaTeX 代码表示结果。

\documentclass{article}%
\usepackage{sagetex}
\begin{document}
\begin{sagesilent}
x = var('x')
Problems = r"Factor each quadratic:\\ \begin{enumerate}"
for i in range(0,20):
    r1 = Integer(randint(-10,10))
    r2 = Integer(randint(-10,10))
    while r1 == -r2:
        r2 = Integer(randint(-10,10))
    poly = expand((x-r1)*(x-r2))
    Problems += r"\item $%s = %s$"%(latex(poly),latex(factor(poly)))
Problems += r"\end{enumerate}"
\end{sagesilent}
\sagestr{Problems}
\end{document}

输出显示在SagemathCloud 中运行：

习惯使用 Sage 和 sagemath 软件包可以让各种数学工作表变得更容易——Sage 可处理所有级别的数学您需要在计算机上本地安装 Sage 或免费SagemathCloud 帐户。

Question 3

为了保险起见，这里有一个纯基于 LuaLaTeX 的解决方案。除了这个luacode包之外，不需要其他包来生成所需的输出。

完整的 MWE：

% !TeX program = lualatex
\documentclass{article}
\usepackage{amsmath} % for 'align*' env.

%% Lua-side code
\usepackage{luacode}
\begin{luacode}
function trinom ( n ) -- n: # of iterations (integer)
local f1,f2,g1,g2,s1
for i=1,n do -- Repeat the following code 'n' times

  -- f1 & f2: integers between -10 and 10, excl. 0
  repeat f1=math.random(-10,10) until f1~=0
  repeat f2=math.random(-10,10) until f2~=0 and f2~=-f1

  -- Compute additive and multiplicative factors 
  g1 = f1+f2 
  g2 = string.format("%+i", f1*f2) -- store as string

  -- Retrieve sign ("s1") of g1
  if g1<0 then s1 = "-" else s1 = "+" end
  g1 = tostring(math.abs(g1)) -- convert to abs. value
  -- Special treatment if abs(g1)=="1"
  if g1=="1" then g1 = "" end

  -- Generate output
  tex.sprint( "(x" .. string.format("%+i",f1) .. ")" ..
              "(x" .. string.format("%+i",f2) .. ")" ..
              "&= x^2" .. s1 .. g1 .. "x" .. g2 ) 
  -- Insert line-break directive if not (yet) at last line
  if n>1 and i<n then tex.sprint ( "\\\\" ) end 

end -- end of 'for' loop
end -- end of 'trinom' function
\end{luacode}

%% TeX-side code
\newcommand\ManyTrinomials[1]{\directlua{trinom(#1)}}

\begin{document}
\allowdisplaybreaks
\begin{align*}
\ManyTrinomials{30} % 30 rows
\end{align*}
\end{document}

备注：如果你不想将方程式嵌入到align*环境中，而只是想逐行打印出来，只需将块

     tex.sprint( "(x" .. string.format("%+i",f1) .. ")" ..
                 "(x" .. string.format("%+i",f2) .. ")" ..
                 "&= x^2" .. s1 .. g1 .. "x" .. g2 )

到

     tex.sprint( "(x" .. string.format("%+i",f1) .. ")" ..
                 "(x" .. string.format("%+i",f2) .. ")" ..
                 "= x^2" .. s1 .. g1 .. "x" .. g2 )

即省略符号&。

Answer