FactorSteps - Maple Help

Student[Basics]

 FactorSteps
 generate steps in factoring polynomials

 Calling Sequence Student[Basics][FactorSteps]( expr, variable ) Student[Basics][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.

Package Usage

 • This function is part of the Student[Basics] package, so it can be used in the short form FactorSteps(..) only after executing the command with(Student[Basics]). However, it can always be accessed through the long form of the command by using Student[Basics][FactorSteps](..).

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Basics}\right]\right):$
 > $\mathrm{FactorSteps}\left({x}^{3}+6{x}^{2}+12x+8\right)$
 $\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{\frac{{{x}}^{{2}}}{{2}}}}{\left[{}\right]}& \begin{array}{ccccc}{}& {{x}}^{{2}}& {+}{4}{}{x}& {+}{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(\left[\colorbox[rgb]{1,1,0.631372549019608}{{}}\right]\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| = |}}\left[{}\right]{\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)
 > $\mathrm{FactorSteps}\left({a}^{2}-{b}^{2}\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{•}& {}& \text{1. Difference of squares}\\ {}& {}& \left[{}\right]\end{array}$ (2)
 > $\mathrm{FactorSteps}\left({x}^{2}-x-12\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{▫}& {}& \text{1. Apply the AC Method}\\ {}& \text{◦}& \text{Rewrite in standard form}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& {\text{Look at the coefficients,}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{A}{}{{x}}^{{2}}{+}{B}{}{x}{+}{C}\\ {}& {}& \left[{"A"}{=}{1}{,}{"B"}{=}{-1}{,}{"C"}{=}{-12}\right]\\ {}& \text{◦}& {\text{Find factors of |AC| = |}}\left[{}\right]{\text{| =}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{12}\\ {}& {}& \left\{{1}{,}{2}{,}{3}{,}{4}{,}{6}{,}{12}\right\}\\ {}& \text{◦}& {\text{Find pairs of the above factors, which, when multiplied equal}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{12}\\ {}& {}& \left\{\left[{}\right]{,}\left[{}\right]{,}\left[{}\right]\right\}\\ {}& \text{◦}& {\text{Which pairs of these factors have a}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{difference}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{of B =}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{-1}{\text{? Found:}}\\ {}& {}& \left[{}\right]{=}{-1}\\ {}& \text{◦}& \text{Split the middle term to use above pair}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& {\text{Factor}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{x}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{out of the first pair}}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& {\text{Factor}}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{-4}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{out of the second pair}}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& {x}{+}{3}\phantom{\rule[-0.0ex]{1.0thickmathspace}{0.0ex}}{\text{is a common factor}}\\ {}& {}& \left[{}\right]\\ {}& \text{◦}& \text{Group common factor}\\ {}& {}& \left[{}\right]\\ {}& {}& \text{This gives:}\\ {}& {}& \left[{}\right]\end{array}$ (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(\left[\colorbox[rgb]{1,1,0.635294117647059}{{}}\right]\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| = |}}\left[{}\right]{\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.