GCD and Polynomials

Question

Is there a simple command to find the greatest common divisor (GCD) of any set of terms?

I am trying to use random numbers to generate complex trinomials of the form ax^2+bx+c such that a,b,c are all integers and a is not one.

But sometimes I'm getting output such as:

5x^2+15x+10 in which case the student would factor out a 5 and do

5(x^2+3x+2)

5(x+2)(x+1)

I'd like my initial complex trinomials to lack a non-one GCD. Even better, if it's not too hard, I'd like to have the option of whether or not non-one GCD is permissible and to show the fully factored form of that trinomial.

The code below doesn't cut it.

What else do I need to do?

p.s. Ironically, I do not know how to show LaTeX output in these forums. Is that possible?

\documentclass{article}

\usepackage{ifthen}
\usepackage{pgf}
 \pgfmathsetseed{\number\pdfrandomseed}
\usepackage{pgffor}

\newcommand{\InitVariables}
{%
\pgfmathsetmacro{\a}{int(random(2,5))}
\pgfmathsetmacro{\b}{int(random(1,5))}
\pgfmathsetmacro{\c}{int(random(1,5))}
\pgfmathsetmacro{\d}{int(random(1,5))}
\pgfmathsetmacro{\A}{int(\a*\c)}
\pgfmathsetmacro{\B}{int(\a*\d+\b*\c)}
\pgfmathsetmacro{\C}{int(\b*\d)}
}

\newcommand{\ComplexTrinomial}
{%
\InitVariables
\large
\(\A{x}^2+\B{x}+\C=\)\hspace{4cm}\((\a{x}+\b)(\c{x}+\d)\)
\vspace{1cm}
}

\newcommand{\MyTrinomials}[1]
{\foreach \x in {1,2,...,#1} {\ComplexTrinomial\\}}

\begin{document}

\MyTrinomials{20}

\end{document}

Answer 1

Randomly generated polynomials? GCD of coefficients of the polynomial? You're using enough mathematics to consider using the sagemath package, located here on CTAN which allows you to harness the power of a free computer algebra system. After setting up the ring of polynomials with integer coefficients, poly1 = R.random_element(2) creates a random polynomial of degree at most 2. The next line checks that the polynomial has degree equal to 2, GCD of the coefficients (called the content) equals 1 (so no common factor) and makes sure the polynomial is not irreducible (can be factored). Note that Sage does the factoring for you so there shouldn't be any mistakes. The information on polynomial commands can be found here.

\documentclass{article}
\usepackage{sagetex}
\usepackage{amsmath}
\begin{document}
\begin{sagesilent}
R.<x>=ZZ[]  #ring of polynomials with integral coefficients
poly1 = R.random_element(2) #choose polynomial from the ring with degree at most 2
while poly1.degree() != 2 or poly1.content() != 1 or   poly1.is_irreducible() == True:
    poly1 = R.random_element(2)
answer1 = factor(poly1)
# while degree is not equal to 2 or the GCD of the coefficients is not 1, or
# the polynomial is irreducible, keep picking a polynomial
\end{sagesilent}
\begin{enumerate}
\item Factor the polynomial $\sage{poly1}$.\\\\\\
\noindent The answer is $\sage{answer1}$\\\\
\begin{sagesilent}
R.<x>=ZZ[]
poly2 = R.random_element(2)
while poly2.degree() != 2 or poly2.content() != 1 or poly2.is_irreducible() == True:
    poly2 = R.random_element(2)
answer2 = factor(poly2)
\end{sagesilent}
\item Factor the polynomial $\sage{poly2}$.\\\\\\
\noindent The answer is $\sage{answer2}$\\\\
\end{enumerate}
\end{document}

Running in Sagemath Cloud gives the output: Using the sagemath package requires Sage on your computer (a pain) OR a free Sagemath Cloud account.

Answer 2

More proof of concept than anything else. It could be written in pure TeX, but I was lazy. Requires LuaLaTeX.

\RequirePackage{luatex85}
\documentclass[varwidth,border=5]{standalone}
\usepackage{amsmath,luacode}
\textwidth=8cm
\begin{luacode*}
function gcd(u, v)
  u = math.abs(u)
  v = math.abs(v)
  if u == v then
    return u
  end
  if u == 0 then
    return v
  end
  if v == 0 then
    return u
  end
  if u % 2 == 0 then
    if v % 2 == 1 then
      return gcd(math.floor(u / 2), v)
    else
      return gcd(math.floor(u / 2), math.floor(v / 2)) * 2
    end
  else
    if v % 2 == 0 then
      return gcd(u, math.floor(v / 2))
    end

    if u > v then
      return gcd(math.floor((u - v) / 2), v)
    else
      return gcd(math.floor((v - u) / 2), u)
    end
  end
end

function solveQuadratic(a, b, c)
  local d, x1, x2;
  d  = b * b - 4 * a * c
  if d >= 0 then
     x1 = (-b - math.sqrt(d)) / (2 * a)
     x2 = (-b + math.sqrt(d)) / (2 * a)
     if x1 ~= math.floor(x1) or x2~= math.floor(x2) then
       x1 = nil
       x2 = nil
     end
  else
    x1 = nil
    x2 = nil
  end
  return {x1 = x1, x2 = x2}
end

function coef(a)
  if a == 1 then
    return ''
  else
    return a
  end
end

function formatCoef(a, c)
  local str = ''
  if a ~= 0 then
    if math.abs(a) > 1 or c == '' then
      str = str .. math.abs(a)
    end
    if a < 0 then
      str = '-' .. str
    end 
    str = str .. c
  end
  return str
end

function formatQuadratic(a, b, c)
  local str = ''
  str = formatCoef(a, 'x^2')
  if str ~= '' and b > 0 then
    str = str .. '+'
  end
  str = str .. formatCoef(b, 'x')
  if str ~= '' and c > 0 then
    str = str .. '+'
  end
  str = str .. formatCoef(c, '')
  return str
end

function formatFactoredQuadratic(x1, x2, g)
  local str = ''
  if x1 == 0 then
    str = 'x'
  else
    if x1 > 0 then
      str = str .. '(x + ' .. x1 .. ')'
    else
      str = str .. '(x ' .. x1 .. ')'
    end
  end
  if x2 == 0 then
    str = str .. 'x'
  else
    if x2 > 0 then
      str = str .. '(x + ' .. x2 .. ')'
    else
      str = str .. '(x ' .. x2 .. ')'
    end
  end
  if g ~= nil then
    if math.abs(g) > 1 then
      str = g .. str
    elseif g == -1 then
      str = '-' .. str
    end
  end
  return str
end
function genEq(n, gc)
  local a, b, c, g, h1, h2, eqs
  if gc == nil then
    h1 = 1
    h2 = 1000 -- some large number
  else
    h1 = 0
    h2 = 1
  end
  eqs = {n=0}
  for a = -n,n do  
    for b = -n,n do
      for c = -n,n do
        if a ~= 0 and b ~=0 and c ~= 0 then
          g = gcd(gcd(a, b), c)
          if g > h1 and g <= h2 then
            if a < 0 then
              s = solveQuadratic(-a, -b, -c)
              g = -g
            else
              s = solveQuadratic(a, b, c)
            end
            if s.x1 ~= nil and s.x2 ~= nil then
              table.insert(eqs, {eq = formatQuadratic(a, b, c),
                fct = formatFactoredQuadratic(-s.x1, -s.x2, g)})
              eqs.n = eqs.n + 1
            end
          end
        end
      end
    end
  end
  return eqs
end
\end{luacode*}
\begin{document}
\begin{align*}
\intertext{Lacking a non-1 GCD}
\directlua{%
eqs = genEq(10,1)
for i = 1,5 do
x = math.random(1, eqs.n)
tex.print(eqs[x].eq .. ' &= ' .. eqs[x].fct ..'\noexpand\\\noexpand\\')
end
}
\\\intertext{Not lacking a non-1 GCD}
\directlua{%
eqs = genEq(10)
for i = 1,5 do
x = math.random(1, eqs.n)
tex.print(eqs[x].eq .. ' &= ' .. eqs[x].fct ..'\noexpand\\\noexpand\\')
end
}
\\
\end{align*}
\end{document}