(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 8.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 157, 7] NotebookDataLength[ 69319, 2303] NotebookOptionsPosition[ 53364, 1898] NotebookOutlinePosition[ 63824, 2132] CellTagsIndexPosition[ 63781, 2129] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell["Math 421 \[FilledSmallCircle] Fall 2010", "Subsubtitle", CellChangeTimes->{{3.49099480109375*^9, 3.4909948084375*^9}, { 3.490994851046875*^9, 3.490994851515625*^9}}, TextAlignment->Center], Cell[CellGroupData[{ Cell["\<\ The Factor Theorem and a corollary of the Fundamental Theorem of Algebra\ \>", "Subtitle", TextAlignment->Center, TextJustification->0], Cell["14 September 2010", "Subsubtitle", CellChangeTimes->{{3.48893028171875*^9, 3.48893030315625*^9}, { 3.489583451984375*^9, 3.489583455796875*^9}, 3.49028881771875*^9, { 3.4905515529375*^9, 3.490551553125*^9}, 3.490875430171875*^9, 3.490994868765625*^9, {3.49194364325*^9, 3.4919436453125*^9}, { 3.49346050165625*^9, 3.49346050515625*^9}}, TextAlignment->Center, TextJustification->0], Cell["\<\ Copyright \[Copyright] 2006\[Dash]2010 by Murray Eisenberg. All rights \ reserved.\ \>", "SmallText", CellChangeTimes->{{3.4909947660625*^9, 3.490994772265625*^9}}, TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell["Prerequisites", "Section", CellChangeTimes->{{3.491943665734375*^9, 3.49194366821875*^9}}], Cell[CellGroupData[{ Cell[TextData[StyleBox["Mathematica", FontSlant->"Italic"]], "Subsection", CellChangeTimes->{{3.491943676609375*^9, 3.491943680703125*^9}}], Cell[TextData[{ "Aside from having a working ", StyleBox["Mathematica", FontSlant->"Italic"], " system at your disposal and knowing how to type input, how to evaluate an \ Input cell, and how to navigate around a notebook, there are really no \ prerequisites.\nIn fact, working through this notebook is a good way to learn \ some ", StyleBox["Mathematica", FontSlant->"Italic"], " basics and even some more advanced ", StyleBox["Mathematica", FontSlant->"Italic"], " techniques." }], "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.491157910984375*^9, 3.491157928234375*^9}, { 3.491158020265625*^9, 3.491158128125*^9}, {3.491158162796875*^9, 3.491158177421875*^9}, {3.491217366828125*^9, 3.49121736778125*^9}}, ParagraphSpacing->{0.5, 0.}, CellID->20739139], Cell[TextData[{ "All the input shown uses the display form ", StyleBox["\[ImaginaryI]", FontFamily->"Courier"], " obtained by typing ", StyleBox["\[EscapeKey]", FontSize->16], StyleBox["ii", FontFamily->"Courier"], StyleBox["\[EscapeKey]", FontSize->16], ", but everything would work identically if you typed ", StyleBox["I", FontFamily->"Courier"], " instead." }], "Text", CellChangeTimes->{3.48969559478125*^9, 3.49115815171875*^9}, CellID->83290015], Cell[TextData[{ "As you work through this notebook in ", StyleBox["Mathematica", FontSlant->"Italic"], ", if you come across a ", StyleBox["Mathematica", FontSlant->"Italic"], " function you don't understand, try using the ", StyleBox["?", FontFamily->"Courier"], " information command to find out something about it. (And follow the ", StyleBox["\[RightSkeleton]", FontFamily->"Helvetica", FontWeight->"Plain", FontSlant->"Plain"], " hyperlink in it, if any, to Documentation Center help.) For example, to \ learn about ", StyleBox["PolynomialQuotient", FontFamily->"Courier"], ", used in the first section below, evaluate:" }], "Text", CellChangeTimes->{{3.49099489565625*^9, 3.490994947390625*^9}, { 3.491943707828125*^9, 3.49194371846875*^9}}], Cell[BoxData[ RowBox[{"?", "PolynomialQuotient"}]], "Input"], Cell[TextData[{ "David Park\[CloseCurlyQuote]s ", StyleBox["Presentations", FontSlant->"Italic"], " application is ", StyleBox["not", FontWeight->"Bold", FontSlant->"Italic"], " needed for this notebook." }], "Text", CellChangeTimes->{3.48969559478125*^9, 3.491217372484375*^9}, CellID->171441541] }, Closed]], Cell[CellGroupData[{ Cell["Mathematics", "Subsection", CellChangeTimes->{ 3.48969559478125*^9, {3.49115821378125*^9, 3.491158215890625*^9}}, CellID->195536689], Cell["\<\ You need to know about adding and multiplying complex numbers.\ \>", "Text", CellChangeTimes->{ 3.48969559478125*^9, {3.491158227984375*^9, 3.491158362609375*^9}, { 3.491943917015625*^9, 3.491943945578125*^9}}, CellID->79531803] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Clearing a variable", "Section", CellChangeTimes->{{3.491943852703125*^9, 3.4919438540625*^9}, { 3.491943957703125*^9, 3.491943959828125*^9}}], Cell[TextData[{ "Many ", StyleBox["Mathematica", FontSlant->"Italic"], " examples will use the variable ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Make sure no value has been assigned to ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " yet:" }], "Text", CellChangeTimes->{{3.49099536290625*^9, 3.490995373609375*^9}, 3.491936317953125*^9, {3.491936366484375*^9, 3.49193638434375*^9}, { 3.491943865890625*^9, 3.491943872109375*^9}}], Cell[BoxData[ RowBox[{"Clear", "[", "z", "]"}]], "Input", CellChangeTimes->{{3.490995338390625*^9, 3.490995339953125*^9}, 3.491936317953125*^9}] }, Closed]], Cell[CellGroupData[{ Cell["The Division Theorem", "Section", CellChangeTimes->{{3.4910034391875*^9, 3.491003451171875*^9}}], Cell[TextData[{ "You know from arithmetic that you can divide one positive integer ", Cell[BoxData[ FormBox["a", TraditionalForm]], FormatType->"TraditionalForm"], " by another positive integer ", Cell[BoxData[ FormBox["b", TraditionalForm]], FormatType->"TraditionalForm"], " to obtain an integer quotient ", Cell[BoxData[ FormBox["q", TraditionalForm]], FormatType->"TraditionalForm"], " and a remainder integer ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], ". That is, given positive integers ", Cell[BoxData[ FormBox["a", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["b", TraditionalForm]], FormatType->"TraditionalForm"], ", there are unique integers ", Cell[BoxData[ FormBox["q", TraditionalForm]], FormatType->"TraditionalForm"], " and ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " with\n\t", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox["a", "b"], StripOnInput->False, FractionBoxOptions->{AllowScriptLevelChange->False}], "=", RowBox[{"q", "+", StyleBox[ FractionBox["r", "b"], StripOnInput->False, FractionBoxOptions->{AllowScriptLevelChange->False}]}]}], TraditionalForm]], FormatType->"TraditionalForm"], ",\nor, equivalently,\n\t", Cell[BoxData[ FormBox[ RowBox[{"a", "=", RowBox[{ RowBox[{"b", " ", "q"}], " ", "+", " ", "r"}]}], TraditionalForm]], FormatType->"TraditionalForm"], ",\nin each case with\n\t", Cell[BoxData[ FormBox[ RowBox[{"0", "\[LessEqual]", "r", "<", "b"}], TraditionalForm]], FormatType->"TraditionalForm"], ".\nthe latter inequality says that the remainder ", Cell[BoxData[ FormBox["r", TraditionalForm]], FormatType->"TraditionalForm"], " is less than the \[OpenCurlyDoubleQuote]divisor\[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox["b", TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.49100348709375*^9, 3.491003711453125*^9}, { 3.491004117390625*^9, 3.4910041295625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "For example, if you use long division to divide 2356 by 14, you obtain a \ quotient of 168 and a remainder of 11, so that\n\t", Cell[BoxData[ FormBox[ RowBox[{"2356", "=", RowBox[{ RowBox[{"14", " ", "168"}], "+", "11."}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.491003719515625*^9, 3.49100372771875*^9}, { 3.49100379421875*^9, 3.491003800640625*^9}, {3.491003832453125*^9, 3.491003892953125*^9}, {3.491003929234375*^9, 3.4910039333125*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], ": can verify that result for you:" }], "Text", CellChangeTimes->{{3.491003943203125*^9, 3.491003968453125*^9}}], Cell[BoxData[ RowBox[{"2356", "\[Equal]", RowBox[{ RowBox[{"14", " ", "168"}], "+", "3"}]}]], "Input", CellChangeTimes->{{3.49100389634375*^9, 3.491003901765625*^9}}], Cell[TextData[{ "And you can use ", StyleBox["Mathematica", FontSlant->"Italic"], " in the first place to calculate the quotient and remainder:" }], "Text", CellChangeTimes->{{3.491004141140625*^9, 3.491004161515625*^9}, 3.491935611359375*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"Quotient", "[", RowBox[{"2356", ",", "14"}], "]"}], ",", RowBox[{"Mod", "[", RowBox[{"235", ",", "14"}], "]"}]}], "}"}]], "Input", CellChangeTimes->{{3.491003764140625*^9, 3.491003817515625*^9}, 3.49100392234375*^9}], Cell[TextData[{ "Likewise, you can use long division of polynomials to divide one polynomial \ ", Cell[BoxData[ FormBox[ RowBox[{"A", "(", "z", ")"}], TraditionalForm]]], " in a variable ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " by another polynomial ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"B", "(", "z", ")"}], " "}], TraditionalForm]]], "in that variable so as to obtain a quotient ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " and a remainder ", Cell[BoxData[ FormBox[ RowBox[{"R", "(", "z"}], TraditionalForm]]], "), with the remainder having a smaller degree than that of the \ \[OpenCurlyDoubleQuote]divisor\[CloseCurlyDoubleQuote] ", Cell[BoxData[ FormBox[ RowBox[{"B", "(", "z", ")"}], TraditionalForm]]], ". More precisely:" }], "Text", CellChangeTimes->{{3.491004183140625*^9, 3.4910042684375*^9}, { 3.4910043151875*^9, 3.49100438678125*^9}, {3.491004540890625*^9, 3.49100457634375*^9}, 3.491935602109375*^9}], Cell[TextData[{ StyleBox["The Polynomial Division Theorem.", FontWeight->"Bold"], " Let ", Cell[BoxData[ FormBox[ RowBox[{"A", "(", "z", ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"B", "(", "z", ")"}], TraditionalForm]]], " polynomials in the variable ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " with real or complex coefficients and with ", Cell[BoxData[ FormBox[ RowBox[{"B", "(", "z", ")"}], TraditionalForm]]], " not the zero polynomial. Then there are unique polynomials ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"R", "(", "z", ")"}], TraditionalForm]]], " in ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " with real or complex coefficients, respectively, such that \n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"A", "(", "z", ")"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"B", "(", "z", ")"}], RowBox[{"Q", "(", "z", ")"}]}], "+", RowBox[{"R", "(", "z", ")"}]}]}], TraditionalForm]]], "\nand\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"deg", " ", RowBox[{"R", "(", "z", ")"}]}], "<", RowBox[{"deg", " ", RowBox[{"B", "(", "z", ")"}]}]}], TraditionalForm]]], "." }], "BlueComments", CellFrame->True, CellChangeTimes->{{3.49099502584375*^9, 3.490995028859375*^9}, { 3.49099511221875*^9, 3.49099512665625*^9}, {3.49100428396875*^9, 3.49100429428125*^9}, {3.491004403421875*^9, 3.491004545625*^9}, { 3.491006038921875*^9, 3.491006050421875*^9}, {3.4913222161875*^9, 3.491322237765625*^9}, {3.491322456234375*^9, 3.4913224646875*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "For example,\n\t", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["z", "4"], "-", RowBox[{"4", SuperscriptBox["z", "3"]}], "+", "z", "+", "6"}], RowBox[{ SuperscriptBox["z", "3"], "-", "2"}]], "=", RowBox[{ RowBox[{"(", RowBox[{"z", "-", "4"}], ")"}], "+", FractionBox[ RowBox[{ RowBox[{"3", "z"}], "-", "2"}], RowBox[{ SuperscriptBox["z", "3"], "-", "2"}]]}]}], StripOnInput->False, FractionBoxOptions->{AllowScriptLevelChange->False}], TraditionalForm]], FormatType->"TraditionalForm"], ",\nor equivalently,\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox["z", "4"], "-", RowBox[{"4", SuperscriptBox["z", "3"]}], "+", "z", "+", "6"}], "=", RowBox[{Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["z", "3"], "-", "2"}], ")"}], RowBox[{"(", RowBox[{"z", "-", "4"}], ")"}]}], "+", RowBox[{"(", RowBox[{ RowBox[{"3", "z"}], "-", "2"}], ")"}]}], TraditionalForm]], FormatType->"TraditionalForm"]]], "."}]}], TraditionalForm]], FormatType->"TraditionalForm"] }], "Text", CellChangeTimes->{{3.49100459878125*^9, 3.4910046225625*^9}, { 3.49100501903125*^9, 3.491005035484375*^9}, {3.491005086359375*^9, 3.491005113359375*^9}, {3.49100515446875*^9, 3.4910052615625*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ "Again, ", StyleBox["Mathematica", FontSlant->"Italic"], " can verify that result\[Ellipsis]" }], "Text", CellChangeTimes->{{3.49100526546875*^9, 3.491005291*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["z", "4"], "-", RowBox[{"4", SuperscriptBox["z", "3"]}], "+", "z", "+", "6"}], "\[Equal]", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ SuperscriptBox["z", "3"], "-", "2"}], ")"}], RowBox[{"(", RowBox[{"z", "-", "4"}], ")"}]}], "+", RowBox[{"(", RowBox[{ RowBox[{"3", "z"}], "-", "2"}], ")"}]}]}], "//", "Expand"}]], "Input", CellChangeTimes->{{3.49100531278125*^9, 3.4910053471875*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{{3.49100533978125*^9, 3.4910053475625*^9}, 3.491936640515625*^9}] }, Closed]], Cell[TextData[{ "\[Ellipsis]and ", StyleBox["Mathematica", FontSlant->"Italic"], " can calculate the polynomial quotient and remainder for you:" }], "Text", CellChangeTimes->{{3.49100535625*^9, 3.491005375828125*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{"a", "[", "z_", "]"}], ":=", RowBox[{ SuperscriptBox["z", "4"], "-", RowBox[{"4", SuperscriptBox["z", "3"]}], "+", "z", "+", "6"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"b", "[", "z", "]"}], ":=", RowBox[{ SuperscriptBox["z", "3"], "-", "2"}]}], ";"}], "\n", RowBox[{"{", RowBox[{ RowBox[{"PolynomialQuotient", "[", RowBox[{ RowBox[{"a", "[", "z", "]"}], ",", RowBox[{"b", "[", "z", "]"}], ",", "z"}], "]"}], ",", RowBox[{"PolynomialRemainder", "[", RowBox[{ RowBox[{"a", "[", "z", "]"}], ",", RowBox[{"b", "[", "z", "]"}], ",", "z"}], "]"}]}], "}"}]}], "Input", CellChangeTimes->{{3.491004934421875*^9, 3.49100500675*^9}, { 3.491005387609375*^9, 3.491005420921875*^9}, 3.49100545365625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"-", "4"}], "+", "z"}], ",", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "z"}]}]}], "}"}]], "Output", CellChangeTimes->{ 3.49100492828125*^9, {3.49100497725*^9, 3.491005009234375*^9}, { 3.491005434078125*^9, 3.491005455015625*^9}, 3.491936643140625*^9}] }, Closed]], Cell[TextData[{ "Probably you\[CloseCurlyQuote]ve only used polynomial long division when \ both ", Cell[BoxData[ FormBox[ RowBox[{"A", "(", "z", ")"}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{"B", "(", "z", ")"}], TraditionalForm]]], " have integer coefficients. In that case, the result still \ holds\[LongDash]and both the quotient ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " and remainder ", Cell[BoxData[ FormBox[ RowBox[{"R", "(", "z", ")"}], TraditionalForm]]], " also have integer coefficients\[LongDash]provided that the coefficient of \ the highest power of ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " in the divisor ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"B", "(", "z", ")"}], Cell[""]}], TraditionalForm]]], " is 1." }], "Text", CellChangeTimes->{{3.491005571484375*^9, 3.49100587859375*^9}, { 3.491322260453125*^9, 3.49132243384375*^9}, 3.491935598515625*^9}], Cell["\<\ A rigorous proof of the theorem, which is beyond the scope of this course, \ uses mathematical induction. The informal idea of the proof is what happens \ in long division of polynomials: at each step the degree of the remainder at \ that step has a lesser degree than does the remainder at the previous step, \ so that you can keep going until you reach a step where the degree of the \ remainder at that step is less than the degree of the divisor.\ \>", "Text", CellChangeTimes->{{3.491005913546875*^9, 3.49100593603125*^9}, { 3.491006101203125*^9, 3.491006459859375*^9}, {3.491935617078125*^9, 3.491935657625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " With paper and pencil, carry out long division to express the given \ polynomial ", Cell[BoxData[ FormBox[ RowBox[{"A", "(", "z", ")"}], TraditionalForm]]], " in the form ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{"B", "(", "z", ")"}], RowBox[{"Q", "(", "z", ")"}]}], "+", RowBox[{"R", "(", "z", ")"}]}], TraditionalForm]]], " for the given polynomial ", Cell[BoxData[ FormBox[ RowBox[{"B", "(", "z", ")"}], TraditionalForm]]], ". Then use ", StyleBox["Mathematica", FontSlant->"Italic"], " to verify the result.\n\t(a) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ SuperscriptBox["z", "3"], "-", RowBox[{"5", SuperscriptBox["z", "2"]}], "-", RowBox[{"4", "z"}], "+", "20"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Q", "(", "z", ")"}], "=", RowBox[{"z", "-", "5"}]}], TraditionalForm]]], ".\n\t(b) ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ SuperscriptBox["z", "3"], "-", RowBox[{"5", SuperscriptBox["z", "2"]}], "-", RowBox[{"4", "z"}], "+", "20"}]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Q", "(", "z", ")"}], "=", RowBox[{"z", "-", "1"}]}], TraditionalForm]]], ".\n\t(b)", Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ SuperscriptBox["z", "5"], "-", RowBox[{"12", SuperscriptBox["z", "4"]}], "+", RowBox[{"48", SuperscriptBox["z", "3"]}], "-", RowBox[{"62", SuperscriptBox["z", "2"]}], "-", RowBox[{"33", "z"}], "+", "90", " "}]}]}], TraditionalForm]]], "and ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Q", "(", "z", ")"}], "=", RowBox[{ SuperscriptBox["z", "2"], "-", RowBox[{"3", "z"}], "+", "1"}]}], TraditionalForm]]], "." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.491935727109375*^9, 3.491935949203125*^9}, {3.491936036453125*^9, 3.491936051125*^9}, { 3.49193609665625*^9, 3.491936172859375*^9}, {3.491936241421875*^9, 3.491936252078125*^9}, {3.49346041234375*^9, 3.4934604205625*^9}, { 3.493460458609375*^9, 3.49346046096875*^9}}, TextJustification->0., ParagraphSpacing->{0.5, 0}] }, Closed]], Cell[CellGroupData[{ Cell["The Factor Theorem", "Section"], Cell["The theorem is:", "Text"], Cell[TextData[{ StyleBox["The Factor Theorem.", FontWeight->"Bold"], " Let ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], " be a polynomial in ", Cell[BoxData[ FormBox["z", TraditionalForm]]], " (with real or complex coefficients) of degree ", Cell[BoxData[ FormBox[ RowBox[{"n", ">", "0"}], TraditionalForm]]], ". Then a (real or complex) number ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]]], " is a root of ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], " if and only if \n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], " ", "=", " ", RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], " ", RowBox[{"Q", "(", "z", ")"}]}]}], TraditionalForm]]], "\nfor some polynomial ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " of degree ", Cell[BoxData[ FormBox[ RowBox[{"n", "-", "1"}], TraditionalForm]]], "." }], "BlueComments", CellFrame->True, CellChangeTimes->{{3.49099502584375*^9, 3.490995028859375*^9}, { 3.49099511221875*^9, 3.49099512665625*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ StyleBox["Proof.", FontWeight->"Bold"], " First assume that ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]]], " is a root of ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], ". By long division, there is a quotient polynomial ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " and a remainder polynomial ", Cell[BoxData[ FormBox[ RowBox[{"R", "(", "z", ")"}], TraditionalForm]]], " for which\n \t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], RowBox[{"Q", "(", "z", ")"}]}], "+", RowBox[{"R", "(", "z", ")"}]}]}], TraditionalForm]]], ".\nSince the divisor ", Cell[BoxData[ FormBox[ RowBox[{"z", " ", "-", " ", SubscriptBox["z", "0"]}], TraditionalForm]]], " is of degree 1, the remainder polynomial ", Cell[BoxData[ FormBox[ RowBox[{"R", "(", "z", ")"}], TraditionalForm]]], " is of degree 0, that is, a constant ", Cell[BoxData[ FormBox["r", TraditionalForm]]], ". Then\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], " ", RowBox[{"Q", "(", "z", ")"}]}], "+", "r"}]}], TraditionalForm]]], ".\nTake ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", SubscriptBox["z", "0"]}], TraditionalForm]]], " in both sides of this equation to obtain \n\t", Cell[BoxData[ FormBox[ RowBox[{"0", "=", RowBox[{ RowBox[{"P", "(", SubscriptBox["z", "0"], ")"}], "=", RowBox[{ RowBox[{"0", " ", RowBox[{"Q", "(", SubscriptBox["z", "0"], ")"}]}], "+", "r"}]}]}], TraditionalForm]]], ", \nso that ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", "0"}], TraditionalForm]]], ". Hence\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], RowBox[{"Q", "(", "z", ")"}]}]}], TraditionalForm]]], ".\nSince ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], " has degree ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " whereas ", Cell[BoxData[ FormBox[ RowBox[{"z", "-", SubscriptBox["z", "0"]}], TraditionalForm]]], " has degree 1, then ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " has degree ", Cell[BoxData[ FormBox[ RowBox[{"n", "-", "1"}], TraditionalForm]]], ".\nConversely, assume that ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], RowBox[{"Q", "(", "z", ")"}]}]}], TraditionalForm]]], " for a polynomial ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " of degree ", Cell[BoxData[ FormBox[ RowBox[{"n", "-", "1"}], TraditionalForm]]], ". Take ", Cell[BoxData[ FormBox[ RowBox[{"z", "=", SubscriptBox["z", "0"]}], TraditionalForm]]], " in this equation to obtain ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", SubscriptBox["z", "0"], ")"}], "=", RowBox[{ RowBox[{"0", " ", RowBox[{"Q", "(", SubscriptBox["z", "0"], ")"}]}], "=", "0"}]}], TraditionalForm]]], ". Hence ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]]], " is a root of ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], ". \[FilledRectangle]" }], "Text", CellChangeTimes->{{3.49099503559375*^9, 3.49099526503125*^9}, { 3.491936439078125*^9, 3.491936442703125*^9}}, ParagraphSpacing->{0.5, 0}], Cell[CellGroupData[{ Cell["Example of the Factor Theorem", "Subsection"], Cell[TextData[{ "Make sure no value has been assigned to the variable ", StyleBox["z", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " yet:" }], "Text", CellChangeTimes->{{3.49099536290625*^9, 3.490995373609375*^9}, 3.491936317953125*^9}], Cell[BoxData[ RowBox[{"Clear", "[", "z", "]"}]], "Input", CellChangeTimes->{{3.490995338390625*^9, 3.490995339953125*^9}, 3.491936317953125*^9}], Cell["Define a polynomial:", "Text", CellChangeTimes->{{3.4909953806875*^9, 3.490995387140625*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"p", "[", "z_", "]"}], ":=", RowBox[{ SuperscriptBox["z", "3"], "-", RowBox[{"5", SuperscriptBox["z", "2"]}], "+", RowBox[{"17", "z"}], "-", "13"}]}]], "Input", CellChangeTimes->{ 3.49099526903125*^9, {3.490996780140625*^9, 3.490996796796875*^9}, { 3.49193666003125*^9, 3.491936680890625*^9}}], Cell["By inspection, 1 is a root of this polynomial:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"p", "[", "1", "]"}]], "Input"], Cell[BoxData["0"], "Output", CellChangeTimes->{3.4909952793125*^9, 3.491936692921875*^9}] }, Closed]], Cell["Give this root a name:", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ SubscriptBox["z", "0"], "=", "1"}], ";"}]], "Input", CellChangeTimes->{{3.49099528315625*^9, 3.490995315703125*^9}}], Cell[TextData[{ "The question is how to form the quotient polynomial ", Cell[BoxData[ FormBox[ RowBox[{"Q", "(", "z", ")"}], TraditionalForm]]], " in ", StyleBox["Mathematica. ", FontSlant->"Italic"], StyleBox["(As usual, begin user-defined ", FontWeight->"Plain", FontSlant->"Plain"], StyleBox["Mathematica", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" names with use lower-case letters.)", FontWeight->"Plain", FontSlant->"Plain"] }], "Text", CellChangeTimes->{3.491936408203125*^9}], Cell["One way to obtain the quotient:", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"q", "[", "z_", "]"}], ":=", RowBox[{"PolynomialQuotient", "[", RowBox[{ RowBox[{"p", "[", "z", "]"}], ",", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ",", "z"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"q", "[", "z", "]"}]}], "Input", CellChangeTimes->{{3.490995292828125*^9, 3.49099532534375*^9}}], Cell[BoxData[ RowBox[{"13", "-", RowBox[{"4", " ", "z"}], "+", SuperscriptBox["z", "2"]}]], "Output", CellChangeTimes->{{3.490995328515625*^9, 3.490995351359375*^9}, { 3.491936543453125*^9, 3.4919365978125*^9}, 3.491936732953125*^9}] }, Closed]], Cell["Another way to obtain the quotient:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Simplify", "[", RowBox[{ RowBox[{"p", "[", "z", "]"}], "/", RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}]}], "]"}]], "Input", CellChangeTimes->{{3.49099541140625*^9, 3.49099541496875*^9}}], Cell[BoxData[ RowBox[{"13", "-", RowBox[{"4", " ", "z"}], "+", SuperscriptBox["z", "2"]}]], "Output", CellChangeTimes->{3.49099541565625*^9, 3.491936746109375*^9}] }, Closed]], Cell["In turn find roots of the quotient:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"quadraticRoots", "=", RowBox[{"Solve", "[", RowBox[{ RowBox[{ RowBox[{"q", "[", "z", "]"}], "\[Equal]", "0"}], ",", "z"}], "]"}]}]], "Input", CellChangeTimes->{{3.490995420765625*^9, 3.49099542846875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"z", "\[Rule]", RowBox[{"2", "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"z", "\[Rule]", RowBox[{"2", "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.490995429734375*^9, 3.491936750453125*^9}] }, Closed]], Cell["Factor the quotient:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Factor", "[", RowBox[{"q", "[", "z", "]"}], "]"}]], "Input"], Cell[BoxData[ RowBox[{"13", "-", RowBox[{"4", " ", "z"}], "+", SuperscriptBox["z", "2"]}]], "Output", CellChangeTimes->{3.4909954353125*^9, 3.491936760484375*^9}] }, Closed]], Cell[TextData[{ "Ugh! That does nothing. You have to tell ", StyleBox["Mathematica", FontSlant->"Italic"], " that you\[CloseCurlyQuote]re looking for factoring \ \[OpenCurlyDoubleQuote]over the complex numbers\[CloseCurlyDoubleQuote]. And \ you may do that as follows, using the ", StyleBox["Extension", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " option to ", StyleBox["Factor", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ":" }], "Text", CellChangeTimes->{ 3.49099543971875*^9, {3.490995506078125*^9, 3.490995562*^9}, { 3.490995700921875*^9, 3.49099570759375*^9}, {3.49193683921875*^9, 3.491936845375*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"factoredQuotient", "=", RowBox[{"Factor", "[", RowBox[{ RowBox[{"q", "[", "z", "]"}], ",", RowBox[{"Extension", "\[Rule]", RowBox[{"{", "\[ImaginaryI]", "}"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.49193653859375*^9, 3.491936540046875*^9}, { 3.49193676703125*^9, 3.491936811171875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}]}]], "Output", CellChangeTimes->{3.4919367783125*^9, 3.491936811859375*^9}] }, Closed]], Cell[TextData[{ "It so happens that the two complex roots of the quadratic quotient \ polynomial have integers as real and complex parts; in other words, these two \ complex roots belong to the set of ", StyleBox["Gaussian integers", FontWeight->"Bold"], ". Then you could also use ", StyleBox["Factor", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], " but with the option ", StyleBox["GaussianIntegers \[Rule] True", FontFamily->"Courier"], ":" }], "Text", CellChangeTimes->{{3.490996132359375*^9, 3.490996173*^9}, 3.491942526359375*^9}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Factor", "[", RowBox[{ RowBox[{"q", "[", "z", "]"}], ",", RowBox[{"GaussianIntegers", "\[Rule]", "True"}]}], "]"}]], "Input", CellChangeTimes->{{3.490996179375*^9, 3.490996182296875*^9}, { 3.491942576609375*^9, 3.491942589109375*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}]}]], "Output", CellChangeTimes->{3.490996183296875*^9, 3.49194258071875*^9}] }, Closed]], Cell["Factor the original cubic:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"factoredCubic", "=", RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], " ", "factoredQuotient"}]}]], "Input", CellChangeTimes->{{3.490996190640625*^9, 3.490996194953125*^9}, { 3.490996286265625*^9, 3.490996289328125*^9}, {3.49194261696875*^9, 3.49194262178125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"-", "1"}], "+", "z"}], ")"}]}]], "Output", CellChangeTimes->{3.490996197984375*^9, 3.4909962369375*^9, 3.49099628978125*^9, 3.491942623484375*^9}] }, Closed]], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Check that the product of three linear factors really is the original \ polynomial. Rather than looking back at the original cubic polynomial ", StyleBox["p[z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], StyleBox[",", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Plain"], " you should do your checking by evaluating a suitable equation of the form\n\ \t", StyleBox["p[z]==", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "\[Ellipsis]\nand seeing that the result is ", StyleBox["True", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], "." }], "Exercise", CellChangeTimes->{{3.49099633578125*^9, 3.490996569859375*^9}}, ParagraphSpacing->{0.5, 0.}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " Repeat the work that was done with the cubic polynomial ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "3"], "-", RowBox[{"5", SuperscriptBox["z", "2"]}], "+", RowBox[{"17", "z"}], "-", "13"}], TraditionalForm]]], " but now doing all the calculations with paper and pencil. (You\ \[CloseCurlyQuote]ll need to carry out a \[OpenCurlyDoubleQuote]long division\ \[CloseCurlyDoubleQuote] of the cubic by the linear polynomial ", Cell[BoxData[ FormBox[ RowBox[{"z", "-", "1"}], TraditionalForm]]], ".)" }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490996628734375*^9, 3.490996758640625*^9}, {3.491942738203125*^9, 3.4919427485625*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Use ", StyleBox["Mathematica", FontSlant->"Italic"], " to factor the original cubic polynomial all at once." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.49099681565625*^9, 3.490996869609375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " With ", StyleBox["Mathematica", FontSlant->"Italic"], ", repeat for ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "3"], "-", RowBox[{"3", SuperscriptBox["z", "2"]}], "+", "z", "+", "5"}], TraditionalForm]]], " what we did above for ", StyleBox["p[z]", FontFamily->"Courier", FontWeight->"Plain", FontSlant->"Plain"], ". Begin by observing that now ", Cell[BoxData[ FormBox[ RowBox[{"-", "1"}], TraditionalForm]]], " is a root." }], "Exercise", CellChangeTimes->{{3.490996963984375*^9, 3.490996989453125*^9}, { 3.490997061625*^9, 3.490997130015625*^9}, {3.490997160765625*^9, 3.490997184296875*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], ". Repeat the preceding exercise but for ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["z", "3"], "-", RowBox[{"\[ImaginaryI]", " ", SuperscriptBox["z", "2"]}], "+", "z", "-", "\[ImaginaryI]"}], TraditionalForm]]], ". Begin by observing (and checking!) that ", Cell[BoxData[ FormBox["\[ImaginaryI]", TraditionalForm]]], " is a root." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490997199046875*^9, 3.490997273734375*^9}}], Cell[TextData[{ "So far, the example and exercises used only half of the Factor Theorem\ \[LongDash]that if ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]]], " is a root, then ", Cell[BoxData[ FormBox[ RowBox[{"z", "-", SubscriptBox["z", "0"]}], TraditionalForm]]], " is a factor of the polynomial. But there is another half to the Factor \ Theorem, the converse: if ", Cell[BoxData[ FormBox[ RowBox[{"z", "-", SubscriptBox["z", "0"]}], TraditionalForm]]], " is a factor of the polynomial, then ", Cell[BoxData[ FormBox[ SubscriptBox["z", "0"], TraditionalForm]]], " is a root. The following exercise illustrates that half with an example." }], "Text", CellChangeTimes->{{3.490997311453125*^9, 3.490997452609375*^9}}], Cell[TextData[{ StyleBox["Exercise ", FontWeight->"Bold"], StyleBox[ CounterBox["Exercise"], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold"], " In ", StyleBox["Mathematica:", FontSlant->"Italic"], " Define the polynomial ", Cell[BoxData[ FormBox[ RowBox[{"p", "(", "z", ")"}], TraditionalForm]]], " to be ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"z", "+", "2"}], ")"}], RowBox[{"(", RowBox[{ SuperscriptBox["z", "2"], "-", RowBox[{"3", "z"}], "+", "5"}], ")"}]}], TraditionalForm]]], ". Multiply the linear factor and quadratic factors there to obtain an \ unfactored cubic polynomial. Finally, verify that ", Cell[BoxData[ FormBox[ RowBox[{"-", "2"}], TraditionalForm]]], " is a root of this unfactored cubic." }], "Exercise", CellChangeTimes->{ 3.48969559478125*^9, 3.48969563040625*^9, {3.490997491765625*^9, 3.490997702109375*^9}}] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Alternate way to factor, given the roots (", StyleBox["advanced", FontSlant->"Italic"], ")" }], "Subsection", CellChangeTimes->{{3.491942787734375*^9, 3.49194282659375*^9}, { 3.4919429625*^9, 3.4919429651875*^9}}], Cell["\<\ There is another way to form the (unexpanded) product of factors of a \ polynomial if you already know its roots:\ \>", "Text", CellChangeTimes->{{3.4919428335*^9, 3.491942892015625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"quadraticProduct", "=", RowBox[{"Apply", "[", RowBox[{"Times", ",", RowBox[{"z", "-", RowBox[{"(", RowBox[{"z", "/.", "quadraticRoots"}], ")"}]}]}], "]"}]}]], "Input", CellChangeTimes->{{3.4909956548125*^9, 3.490995690125*^9}, { 3.490996208046875*^9, 3.490996211796875*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}]}]], "Output", CellChangeTimes->{{3.49099566203125*^9, 3.490995690875*^9}, 3.49099621415625*^9, 3.491942801859375*^9}] }, Closed]], Cell["\<\ Let\[CloseCurlyQuote]s analyze how that worked, step-by-step. Start with:\ \>", "Text", CellChangeTimes->{{3.49099572553125*^9, 3.49099573471875*^9}, { 3.4909957936875*^9, 3.49099582290625*^9}}], Cell[CellGroupData[{ Cell[BoxData["quadraticRoots"], "Input", CellChangeTimes->{{3.4909957429375*^9, 3.4909957531875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"z", "\[Rule]", RowBox[{"2", "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}]}], "}"}], ",", RowBox[{"{", RowBox[{"z", "\[Rule]", RowBox[{"2", "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}]}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4919429074375*^9}] }, Closed]], Cell["\<\ Use the replacement rules in the preceding output to obtain a list of the \ actual roots:\ \>", "Text", CellChangeTimes->{{3.490995834703125*^9, 3.4909958546875*^9}, { 3.491942911390625*^9, 3.49194291634375*^9}, 3.491944963390625*^9}], Cell[BoxData[ RowBox[{"z", "/.", "quadraticRoots"}]], "Input", CellChangeTimes->{{3.490995757265625*^9, 3.490995761171875*^9}}], Cell["\<\ Form the corresponding linear polynomials:\ \>", "Text", CellChangeTimes->{{3.490995879765625*^9, 3.49099589475*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"z", "-", RowBox[{"(", RowBox[{"z", "/.", "quadraticRoots"}], ")"}]}]], "Input", CellChangeTimes->{{3.49099576540625*^9, 3.490995774109375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ",", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}]}], "}"}]], "Output", CellChangeTimes->{3.49099577484375*^9}] }, Closed]], Cell["\<\ Finally, multiply those linear polynomials to obtain the factored quadratic:\ \>", "Text", CellChangeTimes->{{3.490995899453125*^9, 3.490995925640625*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Apply", "[", RowBox[{"Times", ",", RowBox[{"z", "-", RowBox[{"(", RowBox[{"z", "/.", "quadraticRoots"}], ")"}]}]}], "]"}]], "Input", CellChangeTimes->{{3.4909956548125*^9, 3.490995690125*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "-", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}], " ", RowBox[{"(", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"-", "2"}], "+", RowBox[{"3", " ", "\[ImaginaryI]"}]}], ")"}], "+", "z"}], ")"}]}]], "Output", CellChangeTimes->{{3.49099566203125*^9, 3.490995690875*^9}}] }, Closed]], Cell[TextData[{ "That final step used the \[OpenCurlyDoubleQuote]functional programming\ \[CloseCurlyDoubleQuote] construct ", StyleBox["Apply[Times,", FontFamily->"Courier"], StyleBox["\[Ellipsis]", FontFamily->"Times", FontWeight->"Plain", FontSlant->"Italic"], StyleBox["]", FontFamily->"Courier"], " in order to apply the ", StyleBox["Times", FontFamily->"Courier"], " (multiplication) function to the list of two linear polynomials." }], "Text", CellChangeTimes->{{3.4909959785*^9, 3.490996036171875*^9}, 3.49194293696875*^9}], Cell[TextData[{ "Notice that ", StyleBox["Mathematica", FontSlant->"Italic"], " did ", StyleBox["not", FontSlant->"Italic"], " expand\[LongDash]multiply out\[LongDash]the product." }], "Text", CellChangeTimes->{{3.490996249171875*^9, 3.49099632540625*^9}}] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Corollary to the FTA", "Section"], Cell["\<\ The following theorem is very important. Its proof is harder than you might \ at first suppose. One proof uses the theory of integrating a complex-valued \ function of a complex variable around a closed curve in the complex plane.\ \>", "Text", CellChangeTimes->{{3.490997769328125*^9, 3.490997808171875*^9}, 3.491942987328125*^9, {3.491943049375*^9, 3.49194307678125*^9}}], Cell[TextData[{ StyleBox["The Fundamental Theorem of Algebra", FontWeight->"Bold"], " (", StyleBox["FTA", FontWeight->"Bold"], "). Every non-constant polynomial with real or complex coefficients has at \ least one real or complex root." }], "BlueComments", CellFrame->True, CellChangeTimes->{3.49099775865625*^9}], Cell["\<\ Starting with that restatement of the FTA, a proof by mathematical induction \ establishes the following corollary.\ \>", "Text", CellChangeTimes->{{3.491943254921875*^9, 3.491943307546875*^9}, { 3.491943364*^9, 3.4919433668125*^9}}], Cell[TextData[{ StyleBox["Corollary.", FontWeight->"Bold"], " Let ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], " be a non-constant polynomial with complex coefficients, of degree ", Cell[BoxData[ FormBox["n", TraditionalForm]]], ". Then ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], " has exactly ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " (not necessarily distinct) complex roots." }], "BlueComments", CellFrame->True, CellChangeTimes->{{3.49099792696875*^9, 3.490997976140625*^9}, 3.491944045171875*^9}], Cell["\<\ The easiest way to make sense for now of the \"not necessarily distinct\" \ part of the conclusion there is to rephrase the corollary as follows:\ \>", "Text"], Cell[TextData[{ StyleBox["Corollary ", FontWeight->"Bold"], "(restated). Let ", Cell[BoxData[ FormBox[ RowBox[{"P", "(", "z", ")"}], TraditionalForm]]], " be a non-constant polynomial with complex coefficients, of degree ", Cell[BoxData[ FormBox[ RowBox[{"n", " "}], TraditionalForm]]], ". Then there are ", Cell[BoxData[ FormBox["n", TraditionalForm]]], " complex numbers ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["z", "1"], ",", " ", SubscriptBox["z", "2"], ",", " ", "\[Ellipsis]", ",", " ", SubscriptBox["z", "n"]}], TraditionalForm]]], ", not necessarily different from one another, for which\n\t", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"P", "(", "z", ")"}], "=", RowBox[{ RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "0"]}], ")"}], RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "1"]}], ")"}], " ", "\[CenterEllipsis]", " ", RowBox[{"(", RowBox[{"z", "-", SubscriptBox["z", "n"]}], ")"}], Cell[""]}]}], TraditionalForm]]], "." }], "BlueComments", CellFrame->True, CellChangeTimes->{{3.490997961359375*^9, 3.49099799375*^9}, { 3.491944040515625*^9, 3.49194405534375*^9}}, ParagraphSpacing->{0.5, 0}], Cell[TextData[{ "The justification for that restatement is as follows. 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