(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 6.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     23567,        805]
NotebookOptionsPosition[     20332,        706]
NotebookOutlinePosition[     20787,        725]
CellTagsIndexPosition[     20744,        722]
WindowFrame->Normal
ContainsDynamic->False*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Lesson 4    Polynomials", "Section",
 CellChangeTimes->{
  3.461019212487554*^9, {3.461019589561224*^9, 3.4610195898613453`*^9}, 
   3.4612688563987827`*^9}],

Cell["\<\
In this lesson you will learn about some of the built-in functions that can \
be used to manipulate polynomials.\
\>", "Text"],

Cell[CellGroupData[{

Cell["Expanding and Factoring Polynomials and Rational Functions", \
"Subsection"],

Cell[TextData[{
 "So long as ",
 StyleBox["x", "Input"],
 " is undefined,  it can be used as an indeterminant in a polynomial. Here \
are some examples:"
}], "Text",
 CellChangeTimes->{{3.4610019821223717`*^9, 3.4610020608312407`*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", "x", "]"}], "\[IndentingNewLine]", 
 RowBox[{"poly1", "=", 
  RowBox[{
   RowBox[{"x", "^", "3"}], "+", 
   RowBox[{"5", 
    RowBox[{"x", "^", "2"}]}], "+", "3"}]}], "\[IndentingNewLine]", 
 RowBox[{"poly2", "=", 
  RowBox[{
   RowBox[{"x", "^", "2"}], "+", "x", "-", "1"}]}]}], "Input",
 CellChangeTimes->{{3.461002065804661*^9, 3.4610021046120777`*^9}, 
   3.461002184539338*^9, {3.461004334478462*^9, 3.461004338307024*^9}}],

Cell["\<\
Polynomials represented this way behave as they should under addition and \
multiplication.\
\>", "Text",
 CellChangeTimes->{{3.461002114680994*^9, 3.461002165334888*^9}}],

Cell[BoxData[
 RowBox[{"poly1", "+", "poly2"}]], "Input",
 CellChangeTimes->{{3.461002172291005*^9, 3.4610021776261873`*^9}}],

Cell[BoxData[
 RowBox[{"poly1", " ", "poly2"}]], "Input",
 CellChangeTimes->{{3.461002194506895*^9, 3.461002198138068*^9}}],

Cell["\<\
...except that a product is not expanded unless you ask for it.\
\>", "Text",
 CellChangeTimes->{{3.461002239609733*^9, 3.461002290478592*^9}}],

Cell[BoxData[
 RowBox[{"Expand", "[", 
  RowBox[{"poly1", " ", "poly2"}], "]"}]], "Input",
 CellChangeTimes->{{3.461002269196527*^9, 3.461002275603426*^9}}],

Cell["Here are two polynomials in two variables:", "Text",
 CellChangeTimes->{{3.461004376571155*^9, 3.461004397154543*^9}}],

Cell[BoxData[{
 RowBox[{"Clear", "[", "y", "]"}], "\[IndentingNewLine]", 
 RowBox[{
  RowBox[{"expr1", " ", "=", " ", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"1", "-", "x"}], ")"}], "2"], 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"2", "+", 
       RowBox[{"5", "x"}]}], ")"}], "3"]}]}], ";"}], "\n", 
 RowBox[{
  RowBox[{"expr2", " ", "=", " ", 
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"1", "-", "x"}], ")"}], "2"], 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"1", "-", "y"}], ")"}], "4"]}]}], ";"}]}], "Input",
 CellChangeTimes->{{3.4610043702711678`*^9, 3.4610043738984213`*^9}}],

Cell[TextData[{
 "We can expand these expressions completely using ",
 StyleBox["Expand", "Input",
  FontWeight->"Bold"],
 StyleBox[".",
  FontWeight->"Bold"]
}], "Text",
 CellChangeTimes->{{3.461004415727706*^9, 3.4610044189656057`*^9}}],

Cell[BoxData[
 RowBox[{"Expand", "[", "expr1", "]"}]], "Input"],

Cell["\<\
Or we  can expand these with respect to one of the variables.\
\>", "Text",
 CellChangeTimes->{{3.461004428577578*^9, 3.4610044381475*^9}}],

Cell[BoxData[
 RowBox[{"Expand", "[", 
  RowBox[{"expr2", ",", "x"}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"Expand", "[", 
  RowBox[{"expr2", ",", "y"}], "]"}]], "Input"],

Cell[TextData[{
 "In each case, the terms are grouped according to powers of the variable \
used in the ",
 StyleBox["Expand", "Input",
  FontWeight->"Bold"],
 " command."
}], "Text"],

Cell[TextData[{
 "Note however, that ",
 StyleBox["Expand", "Input",
  FontWeight->"Bold"],
 " works only for polynomials. For other types of expressions, you need to \
use other functions. Here are the choices:"
}], "Text"],

Cell[BoxData[
 RowBox[{"?", "*Expand"}]], "Input"],

Cell["\<\
Now let's do the reverse, namely factoring, which works for functions of one \
or several variables.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"Factor", "[", 
  RowBox[{"40", "-", 
   RowBox[{"88", " ", "x"}], "+", 
   RowBox[{"24", " ", 
    SuperscriptBox["x", "2"]}], "-", 
   RowBox[{"21", " ", 
    SuperscriptBox["x", "3"]}], "+", 
   RowBox[{"11", " ", 
    SuperscriptBox["x", "4"]}], "-", 
   RowBox[{"3", " ", 
    SuperscriptBox["x", "5"]}], "+", 
   RowBox[{"2", " ", 
    SuperscriptBox["x", "6"]}]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"Factor", "[", 
  RowBox[{
   RowBox[{"3", " ", 
    SuperscriptBox["x", "2"]}], "+", 
   RowBox[{"3", " ", 
    SuperscriptBox["x", "3"]}], "-", 
   RowBox[{"8", " ", "x", " ", "y"}], "-", 
   RowBox[{"8", " ", 
    SuperscriptBox["x", "2"], " ", "y"}], "-", 
   RowBox[{"3", " ", 
    SuperscriptBox["y", "2"]}], "-", 
   RowBox[{"3", " ", "x", " ", 
    SuperscriptBox["y", "2"]}]}], "]"}]], "Input"],

Cell["\<\
As in the case of expansion, there are several specialized factoring \
functions.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"?", "*Factor"}]], "Input"],

Cell[BoxData[
 RowBox[{"?", "TrigFactor"}]], "Input"],

Cell["\<\
Finally, you can collect all terms according to the power of a certain \
variable.\
\>", "Text"],

Cell[BoxData[
 RowBox[{"expr4", "=", 
  RowBox[{"Expand", "[", 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"1", "+", 
      RowBox[{"2", "x"}], "-", 
      RowBox[{"3", "y"}]}], ")"}], "3"], "]"}]}]], "Input"],

Cell[TextData[{
 "To regroup the terms according to powers of ",
 StyleBox["y", "Input",
  FontWeight->"Bold"],
 ",",
 StyleBox[" ",
  FontWeight->"Bold"],
 "use ",
 StyleBox["Collect", "Input",
  FontWeight->"Bold"],
 ". "
}], "Text"],

Cell[BoxData[
 RowBox[{"Collect", "[", 
  RowBox[{"expr4", ",", "y"}], "]"}]], "Input"],

Cell[TextData[{
 "You can also extract the coefficient of a particular power of ",
 StyleBox["y", "Input",
  FontWeight->"Bold"],
 StyleBox[".",
  FontWeight->"Bold"]
}], "Text",
 CellChangeTimes->{{3.461004519946566*^9, 3.461004527641958*^9}}],

Cell[BoxData[
 RowBox[{"Coefficient", "[", 
  RowBox[{"expr4", ",", "y", ",", "2"}], "]"}]], "Input",
 CellChangeTimes->{{3.4610045440780897`*^9, 3.461004545068012*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Practice", "Subsection",
 CellFrame->{{0, 0}, {0, 2}}],

Cell[TextData[{
 "1) Expand the expression ",
 Cell[BoxData[
  FormBox[
   StyleBox[
    RowBox[{"expr5", "=", 
     RowBox[{
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"x", " ", "+", " ", "y"}], ")"}], "37"], 
      SuperscriptBox[
       RowBox[{"(", 
        RowBox[{"x", "-", "y"}], ")"}], "37"]}]}], "Input"], 
   TraditionalForm]]],
 "."
}], "Text",
 CellChangeTimes->{{3.4479720949295*^9, 3.447972175911315*^9}, {
  3.447972208059383*^9, 3.447972381673978*^9}, {3.447972440728621*^9, 
  3.447972443148983*^9}, {3.460567490848426*^9, 3.460567491434207*^9}}],

Cell["2) Then factor the result.", "Text",
 CellChangeTimes->{{3.4479720949295*^9, 3.447972175911315*^9}, {
  3.447972208059383*^9, 3.447972381673978*^9}, {3.447972440728621*^9, 
  3.447972443148983*^9}, {3.4605674943162613`*^9, 3.460567498185783*^9}}],

Cell[TextData[{
 "3) What is the coefficient of ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["x", "4"], TraditionalForm]], "Input"],
 "in the above? Do you have to ",
 StyleBox["Expand", "Input"],
 " before using ",
 StyleBox["Coefficient", "Input"],
 "?"
}], "Text",
 CellChangeTimes->{{3.4479720949295*^9, 3.447972175911315*^9}, {
  3.447972208059383*^9, 3.447972381673978*^9}, {3.447972440728621*^9, 
  3.447972443148983*^9}, {3.460567500211851*^9, 3.4605675025549173`*^9}}],

Cell[TextData[{
 "4) What is the coefficient of ",
 Cell[BoxData[
  FormBox[
   SuperscriptBox["y", "4"], TraditionalForm]], "Input"],
 "in the above?"
}], "Text",
 CellFrame->{{0, 0}, {2, 0}},
 CellChangeTimes->{{3.4479720949295*^9, 3.447972175911315*^9}, {
   3.447972208059383*^9, 3.447972381673978*^9}, {3.447972440728621*^9, 
   3.447972443148983*^9}, 3.460567506339534*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Rational Functions", "Subsection",
 CellChangeTimes->{{3.460826971890991*^9, 3.460826978145358*^9}}],

Cell["\<\
If you have a rational function, you can expand numerators and denominators \
separately. \
\>", "Text"],

Cell[BoxData[
 RowBox[{"expr3", "=", 
  FractionBox[
   RowBox[{
    RowBox[{"(", 
     RowBox[{"1", "-", 
      RowBox[{"2", "x"}]}], ")"}], 
    RowBox[{"(", 
     RowBox[{"2", "+", "x"}], ")"}]}], 
   RowBox[{
    RowBox[{"(", 
     RowBox[{"1", "-", 
      RowBox[{"3", 
       SuperscriptBox["x", "3"]}]}], ")"}], 
    RowBox[{"(", 
     RowBox[{"1", "+", "x"}], ")"}]}]]}]], "Input",
 CellChangeTimes->{3.4643614410174713`*^9}],

Cell["\<\
The numerator and denominator of such a rational function can be extracted \
using obvious commands:\
\>", "Text",
 CellChangeTimes->{{3.4643613744593477`*^9, 3.464361412309743*^9}}],

Cell[BoxData[{
 RowBox[{"Numerator", "[", "expr3", "]"}], "\[IndentingNewLine]", 
 RowBox[{"Denominator", "[", "expr3", "]"}]}], "Input",
 CellChangeTimes->{{3.464361415480124*^9, 3.464361432055485*^9}}],

Cell["\<\
And the numerator and denominator can be expanded independently:\
\>", "Text",
 CellChangeTimes->{{3.464361448386447*^9, 3.464361469678569*^9}}],

Cell[BoxData[
 RowBox[{"ExpandNumerator", "[", "expr3", "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"ExpandDenominator", "[", "expr3", "]"}]], "Input"],

Cell[TextData[{
 "Using ",
 StyleBox["Expand", "Input",
  FontWeight->"Bold"],
 StyleBox[" ",
  FontWeight->"Bold"],
 "breaks up the rational functions into separate terms, with each term having \
the same denominator."
}], "Text"],

Cell[BoxData[
 RowBox[{"Expand", "[", "expr3", "]"}]], "Input"],

Cell[TextData[{
 " Frequently, we'd prefer a rational function broken up into partial \
fractions, where each term has only one of the  factors of the denominator. \
The relevant ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " function is called ",
 StyleBox["Apart", "Input",
  FontWeight->"Bold"],
 "."
}], "Text",
 CellChangeTimes->{{3.46082749442236*^9, 3.460827500241755*^9}, 
   3.464361518362112*^9, {3.4643618022376347`*^9, 3.464361803339178*^9}}],

Cell[BoxData[
 RowBox[{"Apart", "[", "expr3", "]"}]], "Input"],

Cell[TextData[{
 "The opposite of writing a rational function as a partial fraction expansion \
is putting a sum of rational functions over a common denominator. Here the ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 " function is ",
 StyleBox["Together", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.460826997174938*^9, 3.4608270353646927`*^9}, {
  3.460827358609489*^9, 3.460827405169161*^9}, {3.460827523558076*^9, 
  3.460827541374447*^9}, {3.460827600213331*^9, 3.46082760292653*^9}}],

Cell[BoxData[
 RowBox[{"Together", "[", 
  RowBox[{
   FractionBox["1", 
    RowBox[{"x", "+", "1"}]], "+", 
   FractionBox["1", 
    SuperscriptBox["x", "2"]]}], "]"}]], "Input",
 CellChangeTimes->{{3.460827547946066*^9, 3.460827575567642*^9}, {
  3.4608276157246227`*^9, 3.460827619726635*^9}}],

Cell[BoxData[
 RowBox[{"Apart", "[", "%", "]"}]], "Input",
 CellChangeTimes->{{3.460827626241542*^9, 3.460827630049521*^9}}],

Cell[BoxData[
 RowBox[{"Together", "[", 
  RowBox[{"Apart", "[", "expr3", "]"}], "]"}]], "Input",
 CellChangeTimes->{{3.4608276462781477`*^9, 3.460827660811206*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Simplification of Expressions", "Subsection"],

Cell[TextData[{
 "Factorization can be thought of as a way of simplifying an expression. \
However, there are many possible techniques for simplification. ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 "'s function ",
 StyleBox["Simplify", "Input"],
 " knows many of them:"
}], "Text",
 CellChangeTimes->{{3.460567531444209*^9, 3.4605675393144484`*^9}}],

Cell[BoxData[
 RowBox[{"Simplify", "[", 
  RowBox[{
   SuperscriptBox["x", "x"], 
   SuperscriptBox[
    RowBox[{"(", 
     RowBox[{"1", "-", 
      FractionBox["1", "x"]}], ")"}], "x"]}], "]"}]], "Input"],

Cell[BoxData[
 RowBox[{"Simplify", "[", 
  RowBox[{
   SuperscriptBox[
    RowBox[{"Sin", "[", "x", "]"}], "2"], "+", 
   SuperscriptBox[
    RowBox[{"Cos", "[", "x", "]"}], "2"]}], "]"}]], "Input"],

Cell[TextData[{
 StyleBox["Simplify", "Input",
  FontWeight->"Bold"],
 " encompasses the most common simplification rules, which are applied to the \
expression. ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 "'s idea of the simplest form for an expression may not be the same as \
yours. To apply a larger set of rules to an expression, try ",
 StyleBox["FullSimplify", "Input",
  FontWeight->"Bold"],
 "."
}], "Text",
 CellChangeTimes->{{3.460567586938703*^9, 3.4605676305943003`*^9}}]
}, Open  ]],

Cell[CellGroupData[{

Cell["Homework", "Subsection",
 CellFrame->{{0, 0}, {0, 2}},
 CellChangeTimes->{{3.460825722876402*^9, 3.460825725610227*^9}}],

Cell[TextData[{
 "1) Use ",
 StyleBox["Apart", "Input"],
 " to find partial fraction expansions for the following:"
}], "Text",
 CellChangeTimes->{{3.460825732570158*^9, 3.460825751888042*^9}, 
   3.460825788394679*^9, {3.460827871566867*^9, 3.460827882120472*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"(", "a", ")"}], " ", 
  FractionBox["x", 
   RowBox[{
    SuperscriptBox["x", "3"], "+", 
    RowBox[{"5", "x"}], "+", "6"}]]}]], "Input",
 CellChangeTimes->{{3.447766030616967*^9, 3.447766066835527*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"(", "b", ")"}], " ", 
  FractionBox[
   RowBox[{
    RowBox[{"2", "x"}], "+", "7"}], 
   RowBox[{
    SuperscriptBox["x", "3"], "+", 
    RowBox[{"3", 
     SuperscriptBox["x", "2"]}], "+", 
    RowBox[{"3", "x"}], "+", "1"}]]}]], "Input",
 CellChangeTimes->{{3.447766073806622*^9, 3.447766118892311*^9}}],

Cell[TextData[{
 "2)  How can we extract the \"constant\" term from a polynomial like",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"1", "+", "x"}], ")"}], "3"], 
    SuperscriptBox[
     RowBox[{"(", 
      RowBox[{"1", "+", "y"}], ")"}], "2"]}], TraditionalForm]], "Input"], 
  "Input"],
 "? Warning: The question is ambiguous."
}], "Text",
 CellChangeTimes->{{3.4610048469969378`*^9, 3.461004867227187*^9}, {
  3.4610051745969763`*^9, 3.4610051928344803`*^9}}],

Cell[TextData[{
 "3) Create the table below containing the factorizations of the polynomials \
of the form",
 StyleBox[" ", "Input"],
 StyleBox[Cell[BoxData[
  FormBox[
   RowBox[{
    SuperscriptBox["x", "n"], "-", "1"}], TraditionalForm]], "Input"], 
  "Input"],
 " with ",
 StyleBox["n=1,2,...,10", "Input"],
 "."
}], "Text",
 CellChangeTimes->{{3.461005275422948*^9, 3.461005340230462*^9}, {
  3.461005382828064*^9, 3.461005394411174*^9}}],

Cell[BoxData[
 TagBox[
  TagBox[GridBox[{
     {
      RowBox[{
       RowBox[{"-", "1"}], "+", "x"}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x"}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x", "+", 
         SuperscriptBox["x", "2"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         SuperscriptBox["x", "2"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x", "+", 
         SuperscriptBox["x", "2"], "+", 
         SuperscriptBox["x", "3"], "+", 
         SuperscriptBox["x", "4"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "-", "x", "+", 
         SuperscriptBox["x", "2"]}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x", "+", 
         SuperscriptBox["x", "2"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x", "+", 
         SuperscriptBox["x", "2"], "+", 
         SuperscriptBox["x", "3"], "+", 
         SuperscriptBox["x", "4"], "+", 
         SuperscriptBox["x", "5"], "+", 
         SuperscriptBox["x", "6"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         SuperscriptBox["x", "2"]}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         SuperscriptBox["x", "4"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x", "+", 
         SuperscriptBox["x", "2"]}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", 
         SuperscriptBox["x", "3"], "+", 
         SuperscriptBox["x", "6"]}], ")"}]}]},
     {
      RowBox[{
       RowBox[{"(", 
        RowBox[{
         RowBox[{"-", "1"}], "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x"}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "-", "x", "+", 
         SuperscriptBox["x", "2"], "-", 
         SuperscriptBox["x", "3"], "+", 
         SuperscriptBox["x", "4"]}], ")"}], " ", 
       RowBox[{"(", 
        RowBox[{"1", "+", "x", "+", 
         SuperscriptBox["x", "2"], "+", 
         SuperscriptBox["x", "3"], "+", 
         SuperscriptBox["x", "4"]}], ")"}]}]}
    },
    GridBoxAlignment->{
     "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, 
      "RowsIndexed" -> {}},
    GridBoxSpacings->{"Columns" -> {
        Offset[0.27999999999999997`], {
         Offset[0.5599999999999999]}, 
        Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> {
        Offset[0.2], {
         Offset[0.4]}, 
        Offset[0.2]}, "RowsIndexed" -> {}}],
   Column],
  Function[BoxForm`e$, 
   TableForm[BoxForm`e$]]]], "Output",
 CellChangeTimes->{{3.461005242003171*^9, 3.461005264820322*^9}}],

Cell[TextData[{
 "4) According to ",
 StyleBox["Mathematica",
  FontSlant->"Italic"],
 ", which of these equal expressions is the simplest? That is, what does ",
 StyleBox["Simplify", "Input"],
 " do to them?"
}], "Text",
 CellChangeTimes->{{3.461005477561348*^9, 3.461005521733899*^9}, {
  3.461005784965426*^9, 3.461005791134091*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"(", 
   RowBox[{"x", "+", "b"}], ")"}], "^", "2"}]], "Input",
 CellChangeTimes->{{3.461005647577392*^9, 3.461005652303462*^9}, 
   3.461005820866296*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{"x", "^", "2"}], "+", 
  RowBox[{"2", "b", " ", "x"}], " ", "+", " ", 
  RowBox[{"b", "^", "2"}]}]], "Input",
 CellChangeTimes->{{3.461005730918333*^9, 3.461005735617334*^9}, 
   3.461005841353565*^9, 3.4610060204965057`*^9}],

Cell["What about these expressions?", "Text",
 CellChangeTimes->{{3.4610058029866657`*^9, 3.46100581500467*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"(", 
    RowBox[{"x", "+", "b"}], ")"}], "^", "2"}], "+", "1"}]], "Input",
 CellChangeTimes->{{3.461005647577392*^9, 3.461005652303462*^9}, 
   3.461005820866296*^9, {3.4610058721068068`*^9, 3.461005872558731*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{"x", "^", "2"}], "+", 
  RowBox[{"2", "b", " ", "x"}], " ", "+", " ", 
  RowBox[{"b", "^", "2"}], "+", "1"}]], "Input",
 CellChangeTimes->{{3.461005730918333*^9, 3.461005735617334*^9}, 
   3.461005841353565*^9, {3.4610058746786737`*^9, 3.4610058751078033`*^9}, 
   3.461005927596212*^9}]
}, Open  ]]
}, Open  ]]
},
ScreenStyleEnvironment->"Working",
WindowSize->{772, 645},
WindowMargins->{{114, Automatic}, {Automatic, 0}},
PrintingCopies->1,
PrintingPageRange->{Automatic, Automatic},
ShowSelection->True,
FrontEndVersion->"6.0 for Mac OS X x86 (32-bit) (April 20, 2007)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[590, 23, 163, 3, 67, "Section"],
Cell[756, 28, 136, 3, 26, "Text"],
Cell[CellGroupData[{
Cell[917, 35, 82, 1, 34, "Subsection"],
Cell[1002, 38, 235, 6, 26, "Text"],
Cell[1240, 46, 466, 11, 58, "Input"],
Cell[1709, 59, 181, 4, 26, "Text"],
Cell[1893, 65, 125, 2, 27, "Input"],
Cell[2021, 69, 123, 2, 27, "Input"],
Cell[2147, 73, 153, 3, 26, "Text"],
Cell[2303, 78, 156, 3, 27, "Input"],
Cell[2462, 83, 124, 1, 26, "Text"],
Cell[2589, 86, 654, 21, 64, "Input"],
Cell[3246, 109, 238, 7, 26, "Text"],
Cell[3487, 118, 63, 1, 27, "Input"],
Cell[3553, 121, 149, 3, 26, "Text"],
Cell[3705, 126, 86, 2, 27, "Input"],
Cell[3794, 130, 86, 2, 27, "Input"],
Cell[3883, 134, 183, 6, 26, "Text"],
Cell[4069, 142, 224, 6, 41, "Text"],
Cell[4296, 150, 50, 1, 27, "Input"],
Cell[4349, 153, 124, 3, 26, "Text"],
Cell[4476, 158, 411, 13, 33, "Input"],
Cell[4890, 173, 426, 13, 33, "Input"],
Cell[5319, 188, 105, 3, 26, "Text"],
Cell[5427, 193, 50, 1, 27, "Input"],
Cell[5480, 196, 53, 1, 27, "Input"],
Cell[5536, 199, 106, 3, 26, "Text"],
Cell[5645, 204, 216, 7, 33, "Input"],
Cell[5864, 213, 235, 11, 26, "Text"],
Cell[6102, 226, 87, 2, 27, "Input"],
Cell[6192, 230, 244, 7, 26, "Text"],
Cell[6439, 239, 169, 3, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[6645, 247, 60, 1, 44, "Subsection"],
Cell[6708, 250, 579, 18, 30, "Text"],
Cell[7290, 270, 252, 3, 26, "Text"],
Cell[7545, 275, 479, 13, 30, "Text"],
Cell[8027, 290, 379, 10, 40, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[8443, 305, 106, 1, 34, "Subsection"],
Cell[8552, 308, 114, 3, 26, "Text"],
Cell[8669, 313, 433, 16, 52, "Input"],
Cell[9105, 331, 192, 4, 26, "Text"],
Cell[9300, 337, 203, 3, 43, "Input"],
Cell[9506, 342, 154, 3, 26, "Text"],
Cell[9663, 347, 72, 1, 27, "Input"],
Cell[9738, 350, 74, 1, 27, "Input"],
Cell[9815, 353, 231, 8, 26, "Text"],
Cell[10049, 363, 63, 1, 27, "Input"],
Cell[10115, 366, 462, 12, 41, "Text"],
Cell[10580, 380, 62, 1, 27, "Input"],
Cell[10645, 383, 500, 11, 41, "Text"],
Cell[11148, 396, 296, 8, 47, "Input"],
Cell[11447, 406, 124, 2, 27, "Input"],
Cell[11574, 410, 165, 3, 27, "Input"]
}, Open  ]],
Cell[CellGroupData[{
Cell[11776, 418, 51, 0, 34, "Subsection"],
Cell[11830, 420, 359, 9, 41, "Text"],
Cell[12192, 431, 205, 7, 46, "Input"],
Cell[12400, 440, 198, 6, 33, "Input"],
Cell[12601, 448, 493, 13, 41, "Text"]
}, Open  ]],
Cell[CellGroupData[{
Cell[13131, 466, 126, 2, 44, "Subsection"],
Cell[13260, 470, 265, 6, 26, "Text"],
Cell[13528, 478, 239, 7, 46, "Input"],
Cell[13770, 487, 340, 11, 48, "Input"],
Cell[14113, 500, 544, 16, 29, "Text"],
Cell[14660, 518, 443, 14, 28, "Text"],
Cell[15106, 534, 3703, 122, 213, "Output"],
Cell[18812, 658, 336, 9, 26, "Text"],
Cell[19151, 669, 188, 5, 27, "Input"],
Cell[19342, 676, 259, 6, 27, "Input"],
Cell[19604, 684, 112, 1, 26, "Text"],
Cell[19719, 687, 261, 6, 27, "Input"],
Cell[19983, 695, 321, 7, 27, "Input"]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)