Chapter R - Section R.3 - Polynomials - R.3 Exercises - Page 30: 99

$\displaystyle \begin{array}{ c l } a) & \mathrm{The\ area\ of\ the\ largest\ square\ is} \ ( x+y)^{2}\\ b) & \mathrm{The\ area\ of\ the\ largest\ square\ is} \ x^{2} +y^{2} +xy+xy=x^{2} +2xy+y^{2}\\ c) & \mathrm{The\ expressions\ in\ parts\ } a\ \mathrm{and} \ b\ \mathrm{are\ equivalent\ because\ they\ both\ represent\ the\ same\ area}\\ d) & ( x+y)^{2} =x^{2} +2xy+y^{2} \end{array}$

$\displaystyle \begin{array}{|c|l|} \hline a) & \begin{array}{{>{\displaystyle}l}} \mathrm{The\ formula\ for\ Area\ of\ a\ square\ is} \ A\ =\ s^{2} \ \mathrm{where} \ s \mathrm{\space represents\ the\ length}\\ \mathrm{\ of\ one\ side\ of\ a\ square.\ In\ this\ case,\ define} \ s\ =\ x\ +\ y.\ \mathrm{Then} :\\ A\ =\ ( x\ +\ y)^{2} \end{array}\\ & \\ \hline b) & \begin{array}{{>{\displaystyle}l}} \mathrm{In\ this\ case,\ the\ area\ of\ the\ largest\ square\ is\ the\ sum\ of\ the\ four\ smaller}\\ \mathrm{figures\ having\ the\ same\ area\ as\ the\ largest\ square.} \ \\ \\ \begin{array}{ c l } x^{2} & \mathrm{is\ the\ area\ of\ the\ larger\ blue\ shaded\ squqre}\\ y^{2} & \mathrm{is\ the\ area\ of\ the\ smaller\ blue\ shaded\ square}\\ xy & \mathrm{is\ the\ area\ of\ the\ yellow\ shaded\ rectangle}\\ xy & \mathrm{is\ the\ area\ of\ the\ orange\ shaded\ rectange} \end{array}\\ \\ \mathrm{Summing\ the\ areas\ gives} :\\ \begin{array}{ c l } A & =x^{2} +y^{2}\\ & =x^{2} +2xy+y^{2} \end{array} \end{array}\\ & \\ \hline c) & \begin{array}{{>{\displaystyle}l}} \mathrm{The\ expressions\ in\ parts\ } a\ \mathrm{and} \ b\ \mathrm{are\ equivalent\ because\ they\ both\ represent}\\ \mathrm{\ the\ same\ area} .\mathrm{We\ can\ show\ this\ algebraically\ by\ setting\ the\ expressions\ for}\\ a\ \mathrm{and} \ b\ \mathrm{equal\ to\ each\ other}\\ \\ \begin{array}{ r l l } ( x+y)^{2} & =x^{2} +2xy+y^{2} & \mathrm{Set\ the\ areas\ of} \ a\ \mathrm{and} \ b\ \mathrm{equal}\\ & & \\ ( x+y)( x+y) & =x^{2} +2xy+y^{2} & \mathrm{Expand} \ ( x+y)^{2}\\ & & \\ x( x+y) +y( x+y) & =x^{2} +2xy+y^{2} & \mathrm{Apply\ the\ distributive\ property}\\ & & \\ x^{2} +xy+xy+y^{2} & =x^{2} +2xy+y^{2} & \\ & & \\ x^{2} +2xy+y^{2} & =x^{2} +2xy+y^{2} & \begin{array}{{>{\displaystyle}l}} \mathrm{Both\ sides\ are\ equal,\ therefore}\\ \mathrm{the\ expressions\ must\ be\ equivalent} \end{array} \end{array} \end{array}\\ & \\ \hline d) & ( x+y)^{2} =x^{2} +2xy+y^{2}\\ \hline \end{array}$