(* 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[ 17979, 639] NotebookOptionsPosition[ 15772, 563] NotebookOutlinePosition[ 16230, 582] CellTagsIndexPosition[ 16187, 579] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Lesson 4 Answers - Polynomials", "Section", CellChangeTimes->{ 3.461019212487554*^9, {3.4610193124157667`*^9, 3.461019319964493*^9}, { 3.461268864065989*^9, 3.461268865627747*^9}, {3.465130327301104*^9, 3.465130333026298*^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}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Apart", "[", FractionBox["x", RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"5", "x"}], "+", "6"}]], "]"}]], "Input", CellChangeTimes->{{3.447766170558778*^9, 3.4477661792226887`*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ RowBox[{"-", FractionBox["1", RowBox[{"8", " ", RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}]}]]}], "+", FractionBox[ RowBox[{"6", "+", "x"}], RowBox[{"8", " ", RowBox[{"(", RowBox[{"6", "-", "x", "+", SuperscriptBox["x", "2"]}], ")"}]}]]}]], "Output", CellChangeTimes->{3.447766180016617*^9}, FontSlant->"Italic"] }, Open ]], 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}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Apart", "[", " ", FractionBox[ RowBox[{ RowBox[{"2", "x"}], "+", "7"}], RowBox[{ SuperscriptBox["x", "3"], "+", RowBox[{"3", SuperscriptBox["x", "2"]}], "+", RowBox[{"3", "x"}], "+", "1"}]], "]"}]], "Input", CellChangeTimes->{{3.447766184357296*^9, 3.4477661913597937`*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ FractionBox["5", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "3"]], "+", FractionBox["2", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "2"]]}]], "Output", CellChangeTimes->{3.447766192612523*^9}, FontSlant->"Italic"] }, Open ]], 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[CellGroupData[{ Cell[BoxData[ RowBox[{"poly", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "^", "3"}], RowBox[{ RowBox[{"(", RowBox[{"1", "+", "y"}], ")"}], "^", "2"}]}]}]], "Input", CellChangeTimes->{{3.461004897580017*^9, 3.461004911322681*^9}}, FontSlant->"Italic"], Cell[BoxData[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "x"}], ")"}], "3"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"1", "+", "y"}], ")"}], "2"]}]], "Output", CellChangeTimes->{3.461004913284855*^9}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "If we think of ", StyleBox["poly", "Input"], " as a polynomial in ", StyleBox["x", "Input"], " with coefficients that are polynomials in ", StyleBox["y", "Input"], ", then the constant term is " }], "Text", CellChangeTimes->{{3.461004876448987*^9, 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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[{ 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FontSlant->"Italic"], Cell[BoxData[ SuperscriptBox[ RowBox[{"(", RowBox[{"b", "+", "x"}], ")"}], "2"]], "Output", CellChangeTimes->{{3.461005910315654*^9, 3.461005922552409*^9}}, FontSlant->"Italic"] }, Open ]], Cell[TextData[{ "So ", StyleBox["Mathematica", FontSlant->"Italic"], " thinks that", StyleBox["(x+b)^2", "Input"], " is simpler than ", StyleBox["x^2+2bx+b^2", "Input"], "." }], "Text", CellChangeTimes->{{3.4610059382380943`*^9, 3.461006009566622*^9}, { 3.461006163895192*^9, 3.461006169302556*^9}}, FontSlant->"Italic"], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"Simplify", "[", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"x", "+", "b"}], ")"}], "^", "2"}], "+", "1"}], "]"}], "\[IndentingNewLine]", RowBox[{"Simplify", "[", RowBox[{ RowBox[{"x", "^", "2"}], "+", RowBox[{"2", "b", " ", "x"}], " ", "+", " ", RowBox[{"b", "^", "2"}], "+", "1"}], "]"}]}], "Input", CellChangeTimes->{{3.4610058851785727`*^9, 3.461005920952647*^9}, { 3.461006035962308*^9, 3.461006039371542*^9}}, 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