 FactorSteps - Maple Help

Student[Basics]

 FactorSteps
 generate steps in factoring polynomials Calling Sequence FactorSteps( expr, variable ) FactorSteps( expr, implicitmultiply = true ) Parameters

 expr - string or expression variable - (optional) variable to collect the terms by implicitmultiply - (optional) truefalse output = ... - (optional) displaystyle = ... - (optional) Description

 • The FactorSteps command accepts a polynomial and displays the steps required to factor the expression.
 • If expr is a string, then it is parsed into an expression using InertForm:-Parse so that no automatic simplifications are applied, and thus no steps are missed.
 • The implicitmultiply option is only relevant when expr is a string.  This option is passed directly on to the InertForm:-Parse command and will cause things like 2x to be interpreted as 2*x, but also, xyz to be interpreted as x*y*z.
 • The output and displaystyle options are described in Student:-Basics:-OutputStepsRecord. The return value is controlled by the output option.
 • This function is part of the Student:-Basics package. Examples

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right):$
 > $\mathrm{FactorSteps}\left({x}^{3}+6{x}^{2}+12x+8\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{▫}& {}& \text{1. Trial Evaluations}\\ {}& \text{◦}& \text{Rewrite in standard form}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{The factors of the constant coefficient}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}8\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{are:}\\ {}& {}& {C}{=}\left\{{1}{,}{2}{,}{4}{,}{8}\right\}\\ {}& \text{◦}& \text{Trial evaluations of}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{in}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{±}C\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{find}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{=}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}-2\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{satisfies the equation, so}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x+2\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{is a factor}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Divide by}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x+2\\ {}& {}& \begin{array}{cc}\stackrel{\phantom{{{z}}^{{2}}}}{\left[{}\right]}& \begin{array}{ccccc}{}& \phantom{{\mathrm{PP}}}{{x}}^{{2}}& \phantom{{P}}{+}{4}{}{x}& \phantom{{\mathrm{PP}}}{+}{4}& {}\\ {)}\phantom{{{x}}^{{2}}}& \phantom{{1}}{{x}}^{{3}}& \phantom{{1}}{+}{6}{}{{x}}^{{2}}& \phantom{{1}}{+}{12}{}{x}& \phantom{{1}}{+}{8}\\ {}& \multicolumn{2}{c}{\frac{{{x}}^{{3}}{+}{2}{}{{x}}^{{2}}}{\phantom{{.}}}}& {}\\ {}& {}& \multicolumn{2}{c}{{4}{}{{x}}^{{2}}{+}{12}{}{x}}& {}& {}\\ {}& {}& \multicolumn{2}{c}{\frac{{4}{}{{x}}^{{2}}{+}{8}{}{x}}{\phantom{{.}}}}& {}& {}\\ {}& {}& {}& \multicolumn{2}{c}{{4}{}{x}{+}{8}}& {}& {}& {}\\ {}& {}& {}& \multicolumn{2}{c}{\frac{{4}{}{x}{+}{8}}{\phantom{{.}}}}& {}& {}& {}\\ {}& {}& {}& {}& {0}\hfill & {}& {}& {}& {}\end{array}\end{array}\\ {}& \text{◦}& \text{Quotient times divisor from long division}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{2. Examine term:}\\ {}& {}& {{x}}^{{2}}{+}{4}{}{x}{+}{4}\\ \text{▫}& {}& \text{3. Apply the AC Method}\\ {}& \text{◦}& \text{Examine quadratic}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Look at the coefficients,}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}A{}{x}^{2}+B{}x+C\\ {}& {}& \left[{"A"}{=}{1}{,}{"B"}{=}{4}{,}{"C"}{=}{4}\right]\\ {}& \text{◦}& \text{Find factors of |AC| = |}1\cdot 4\text{| =}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}4\\ {}& {}& \left\{{1}{,}{2}{,}{4}\right\}\\ {}& \text{◦}& \text{Find pairs of the above factors, which, when multiplied equal}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}4\\ {}& {}& \left\{\left[{}\right]{,}\left[{}\right]\right\}\\ {}& \text{◦}& \text{Which pairs of these factors have a}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{sum}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{of B =}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}4\text{? Found:}\\ {}& {}& \left[{}\right]{=}{4}\\ {}& \text{◦}& \text{Split the middle term to use above pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Factor}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{out of the first pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Factor}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}2\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{out of the second pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& x+2\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{is a common factor}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Group common factor}\\ {}& {}& \left[{}\right]\\ {}& {}& \text{This gives:}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{4. This gives:}\\ {}& {}& \left[{}\right]\end{array}$ (1)
 > $\mathrm{FactorSteps}\left({a}^{2}-{b}^{2}\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{•}& {}& \text{1. This is a difference of squares, in the form}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}^{2}-{B}^{2}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{2. Apply difference of squares rule:}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}^{2}-{B}^{2}=\left(A+B\right){}\left(A-B\right)\\ {}& {}& \left[{}\right]\end{array}$ (2)
 > $\mathrm{FactorSteps}\left({x}^{2}-x-12\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{▫}& {}& \text{1. Apply the AC Method}\\ {}& \text{◦}& \text{Rewrite in standard form}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Look at the coefficients,}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}A{}{x}^{2}+B{}x+C\\ {}& {}& \left[{"A"}{=}{1}{,}{"B"}{=}{-1}{,}{"C"}{=}{-12}\right]\\ {}& \text{◦}& \text{Find factors of |AC| = |}1\cdot \left(-12\right)\text{| =}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}12\\ {}& {}& \left\{{1}{,}{2}{,}{3}{,}{4}{,}{6}{,}{12}\right\}\\ {}& \text{◦}& \text{Find pairs of the above factors, which, when multiplied equal}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}12\\ {}& {}& \left\{\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]\right\}\\ {}& \text{◦}& \text{Which pairs of these factors have a}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{difference}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{of B =}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}-1\text{? Found:}\\ {}& {}& \left[{}\right]{=}{-1}\\ {}& \text{◦}& \text{Split the middle term to use above pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Factor}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}x\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{out of the first pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Factor}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}-4\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{out of the second pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& x+3\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{is a common factor}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Group common factor}\\ {}& {}& \left[{}\right]\\ {}& {}& \text{This gives:}\\ {}& {}& \left[{}\right]\end{array}$ (3)
 > $\mathrm{FactorSteps}\left(\frac{2{y}^{2}}{5}+\frac{113y}{5}+33\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{•}& {}& \text{1. Remove rationals and common factor}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{2. Examine term:}\\ {}& {}& \left[{}\right]\\ \text{▫}& {}& \text{3. Apply the AC Method}\\ {}& \text{◦}& \text{Examine quadratic}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Look at the coefficients,}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}A{}{y}^{2}+B{}y+C\\ {}& {}& \left[{"A"}{=}{2}{,}{"B"}{=}{113}{,}{"C"}{=}{165}\right]\\ {}& \text{◦}& \text{Find factors of |AC| = |}2\cdot 165\text{| =}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}330\\ {}& {}& \left\{{1}{,}{2}{,}{3}{,}{5}{,}{6}{,}{10}{,}{11}{,}{15}{,}{22}{,}{30}{,}{33}{,}{55}{,}{66}{,}{110}{,}{165}{,}{330}\right\}\\ {}& \text{◦}& \text{Find pairs of the above factors, which, when multiplied equal}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}330\\ {}& {}& \left\{\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]\right\}\\ {}& \text{◦}& \text{Which pairs of these factors have a}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{sum}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{of B =}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}113\text{? Found:}\\ {}& {}& \left[{}\right]{=}{113}\\ {}& \text{◦}& \text{Split the middle term to use above pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Factor}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}y\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{out of the first pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Factor}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}55\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{out of the second pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& 2{}y+3\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}\text{is a common factor}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Group common factor}\\ {}& {}& \left[{}\right]\\ {}& {}& \text{This gives:}\\ {}& {}& \left[{}\right]\\ \text{•}& {}& \text{4. This gives:}\\ {}& {}& \left[{}\right]\end{array}$ (4) Compatibility

 • The Student[Basics][FactorSteps] command was introduced in Maple 2021.