 Solution Steps - Maple Help

 Solution Steps

Maple 2021 includes numerous new algorithms for showing step-by-step solutions for a variety of problems in mathematics. Long Division

The LongDivision command gives a visual solution to an arithmetic or polynomial long division problem, showing all of the intermediate steps.

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right):$
 > $\mathrm{LongDivision}\left(48{x}^{4}+284{x}^{3}+620{x}^{2}+593x+210,2x+3\right)$
 $\begin{array}{cc}\stackrel{\phantom{\frac{{{x}}^{{2}}}{{2}}}}{{\mathrm{%+}}{}\left({2}{}{x}{,}{3}\right)}& \begin{array}{cccccc}{}& {24}{}{{x}}^{{3}}& {+}{106}{}{{x}}^{{2}}& {+}{151}{}{x}& {+}{70}& {}\\ {)}\phantom{{{x}}^{{2}}}& \phantom{{1}}{48}{}{{x}}^{{4}}& \phantom{{1}}{+}{284}{}{{x}}^{{3}}& \phantom{{1}}{+}{620}{}{{x}}^{{2}}& \phantom{{1}}{+}{593}{}{x}& \phantom{{1}}{+}{210}\\ {}& \multicolumn{2}{c}{\frac{{48}{}{{x}}^{{4}}{+}{72}{}{{x}}^{{3}}}{\phantom{{.}}}}& {}\\ {}& {}& \multicolumn{2}{c}{{212}{}{{x}}^{{3}}{+}{620}{}{{x}}^{{2}}}& {}& {}\\ {}& {}& \multicolumn{2}{c}{\frac{{212}{}{{x}}^{{3}}{+}{318}{}{{x}}^{{2}}}{\phantom{{.}}}}& {}& {}\\ {}& {}& {}& \multicolumn{2}{c}{{302}{}{{x}}^{{2}}{+}{593}{}{x}}& {}& {}& {}\\ {}& {}& {}& \multicolumn{2}{c}{\frac{{302}{}{{x}}^{{2}}{+}{453}{}{x}}{\phantom{{.}}}}& {}& {}& {}\\ {}& {}& {}& {}& \multicolumn{2}{c}{{140}{}{x}{+}{210}}& {}& {}& {}& {}\\ {}& {}& {}& {}& \multicolumn{2}{c}{\frac{{140}{}{x}{+}{210}}{\phantom{{.}}}}& {}& {}& {}& {}\\ {}& {}& {}& {}& {}& {0}\hfill & {}& {}& {}& {}& {}\end{array}\end{array}$ (1.1)
 >
 (1.2) Factoring

The FactorSteps command shows the steps in factoring a polynomial.

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right):$
 > $\mathrm{FactorSteps}\left({x}^{3}+6{x}^{2}+12x+8\right)$
 $\begin{array}{lll}{}& {}& \left[{}\right]\\ \text{â–«}& {}& \text{1. Trial Evaluations}\\ {}& \text{â—¦}& \text{Rewrite in standard form}\\ {}& {}& \left[{}\right]\\ {}& \text{â—¦}& \text{The factors of the constant coefficient}8\text{are:}\\ {}& {}& {C}{=}\left\{{1}{,}{2}{,}{4}{,}{8}\right\}\\ {}& \text{â—¦}& \text{Trial evaluations of}x\text{in}\text{Â±}C\text{find}x\text{=}-2\text{satisfies the equation, so}x+2\text{is a factor}\\ {}& {}& \left[{}\right]\\ {}& \text{â—¦}& \text{Divide by}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{â—¦}& \text{Quotient times divisor from long division}\\ {}& {}& \left[{}\right]\\ \text{â€¢}& {}& \text{2. Examine term:}\\ {}& {}& {{x}}^{{2}}{+}{4}{}{x}{+}{4}\\ \text{â–«}& {}& \text{3. Apply the AC Method}\\ {}& \text{â—¦}& \text{Examine quadratic}\\ {}& {}& \left(\left[\colorbox[rgb]{1,1,0.631372549019608}{{}}\right]\right)\\ {}& \text{â—¦}& \text{Look at the coefficients,}A{}{x}^{2}+B{}x+C\\ {}& {}& \left[{"A"}{=}{1}{,}{"B"}{=}{4}{,}{"C"}{=}{4}\right]\\ {}& \text{â—¦}& \text{Find factors of |AC| = |}1\cdot 4\text{| =}4\\ {}& {}& \left\{{1}{,}{2}{,}{4}\right\}\\ {}& \text{â—¦}& \text{Find pairs of the above factors, which, when multiplied equal}4\\ {}& {}& \left\{\left[{}\right]{,}\left[{}\right]\right\}\\ {}& \text{â—¦}& \text{Which pairs of these factors have a}\text{sum}\text{of B =}4\text{? Found:}\\ {}& {}& \left[{}\right]{=}{4}\\ {}& \text{â—¦}& \text{Split the middle term to use above pair}\\ {}& {}& \left[{}\right]\\ {}& \text{â—¦}& \text{Factor}x\text{out of the first pair}\\ {}& {}& \left[{}\right]\\ {}& \text{â—¦}& \text{Factor}2\text{out of the second pair}\\ {}& {}& \left[{}\right]\\ {}& \text{â—¦}& x+2\text{is a common factor}\\ {}& {}& \left[{}\right]\\ {}& \text{â—¦}& \text{Group common factor}\\ {}& {}& \left[{}\right]\\ {}& {}& \text{This gives:}\\ {}& {}& \left[{}\right]\\ \text{â€¢}& {}& \text{4. This gives:}\\ {}& {}& \left[{}\right]\end{array}$ (2.1) Solve

The SolveSteps command shows the steps in solving an equation or system of equations

 > $\mathrm{with}\left(\mathrm{Student}:-\mathrm{Basics}\right):$
 > $\mathrm{SolveSteps}\left(5{ⅇ}^{4x}=16\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left[{}\right]{=}{16}\\ \text{â–«}& {}& \text{Convert from exponential equation}\\ {}& \text{â—¦}& \text{Divide both sides by}5\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Simplify}\\ {}& {}& \left[{}\right]{=}\frac{{16}}{{5}}\\ {}& \text{â—¦}& \text{Apply ln to each side}\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Apply ln rule: ln(e^b) = b}\\ {}& {}& {4}{}{x}{=}{\mathrm{ln}}{}\left(\frac{{16}}{{5}}\right)\\ \text{â€¢}& {}& \text{Divide both sides by}4\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ \text{â€¢}& {}& \text{Exact solution}\\ {}& {}& {x}{=}\frac{{\mathrm{ln}}{}\left(\frac{{16}}{{5}}\right)}{{4}}\\ \text{â€¢}& {}& \text{Approximate solution}\\ {}& {}& {x}{=}{0.2907877025}\end{array}$ (3.1)
 > $\mathrm{SolveSteps}\left(\left[12x+y=18,7x-8y=32\right]\right)$
 $\begin{array}{lll}{}& {}& \text{Let's solve}\\ {}& {}& \left[\left[{}\right]{=}{18}{,}\left[{}\right]{=}{32}\right]\\ \text{â€¢}& {}& \text{Pick the 2nd}\text{equation to solve for}y\\ {}& {}& \left[{}\right]{=}{32}\\ \text{â–«}& {}& \text{A}\text{: isolate for}y\\ {}& \text{â—¦}& \text{Subtract}7\cdot x\text{from both sides}\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Simplify}\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Divide both sides by}-8\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Simplify}\\ {}& {}& {y}{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Solution}\\ {}& {}& {y}{=}{-}{4}{+}\frac{{7}{}{x}}{{8}}\\ \text{â€¢}& {}& \text{Substitute the value of}y=-4+\frac{7{}x}{8}\text{into the 1st}\text{equation of the system}\\ {}& {}& \left[{}\right]{=}{18}\\ \text{â–«}& {}& \text{Solve for}x\\ {}& \text{â—¦}& \text{Evaluate subtraction and addition}\\ {}& {}& \frac{{103}{}{x}}{{8}}{-}{4}{=}{18}\\ {}& \text{â—¦}& \text{Add}4\text{to both sides}\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Simplify}\\ {}& {}& \left[{}\right]{=}{22}\\ {}& \text{â—¦}& \text{Divide both sides by}\frac{103}{8}\\ {}& {}& \left[{}\right]{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Simplify}\\ {}& {}& {x}{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Rewrite division as multiplication by reciprocal}\\ {}& {}& {x}{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Multiply fraction and reduce by gcd}\\ {}& {}& {x}{=}\frac{{176}}{{103}}\\ \text{â€¢}& {}& \text{Substitute}x=\frac{176}{103}\text{into equation}\text{A}\text{}\\ {}& {}& {y}{=}\left[{}\right]\\ \text{â–«}& {}& \text{Solve for}y\\ {}& \text{â—¦}& \text{Evaluate multiplication and division}\\ {}& {}& {y}{=}\left[{}\right]\\ {}& \text{â—¦}& \text{Evaluate subtraction and addition}\\ {}& {}& {y}{=}{-}\frac{{258}}{{103}}\\ \text{â€¢}& {}& \text{Solution}\\ {}& {}& \left[{x}{=}\frac{{176}}{{103}}{,}{y}{=}{-}\frac{{258}}{{103}}\right]\end{array}$ (3.2) Calculus: Integration, Differentiation, and Limits

The ShowSolution command has been improved to show more detailed steps when solving integration, differentiation, and limit problems.

 >
 > $\mathrm{ShowSolution}\left({\int }\mathrm{sin}{\left(x\right)}^{2}{ⅆ}x\right)$
 $\begin{array}{lll}{}& {}& \text{Integration Steps}\\ {}& {}& {\int }{{\mathrm{sin}}{}\left({x}\right)}^{{2}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\\ \text{â–«}& {}& \text{1. Rewrite}\\ {}& \text{â—¦}& \text{Equivalent expression}\\ {}& {}& {{\mathrm{sin}}{}\left({x}\right)}^{{2}}{=}\frac{{1}}{{2}}{-}\frac{{\mathrm{cos}}{}\left({2}{}{x}\right)}{{2}}\\ {}& {}& \text{This gives:}\\ {}& {}& {\int }\left(\frac{{1}}{{2}}{-}\frac{{\mathrm{cos}}{}\left({2}{}{x}\right)}{{2}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\\ \text{â–«}& {}& \text{2. Apply the}\mathbf{\text{sum}}\text{rule}\\ {}& \text{â—¦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ {}& {}& {\int }\left({f}{}\left({x}\right){+}{g}{}\left({x}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{=}{\int }{f}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}\end{array}$