{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 53 "High School Modul es > Precalculus by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 256 28 "Long Division of Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 74 "\nDivision of polynomials by binomials of degree ones usi ng long division.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Ex ecute the Code Resource section first. Although there will be no outpu t immediately, these definitions are used later in this worksheet.]" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 " 0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "MixedNumber := proc(N, D) \n\nlocal Q, R;\nR := irem(N, D, 'Q');\nprint(N/D = Q,` + `,R/D);\n \+ end proc:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "PolyDivi de := proc(S,T)\n local Q,R;\n Q := quo( S, T, x, 'R'):\n p rint(S/T = Q + R/T);\n end proc:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 436 "LongDiv1 := proc( P, r)\n local A, C, d, i,j,k, rows;\n d := degree( P); rows := 3;\n for k from 0 to d do \+ C||k := coeff(P, x, k) ; od;\n A := array( [seq( [ seq(` `, j = \+ 0..(d+3) ) ], i = 1..rows) ]);\n\n A[3,1] := x-r;\n for k from 0 to d do A[3,d - k + 3] := C||k*x^k ; od;\n for k from 0 to d d o A[2,d - k + 3] := `__`; od;\n for k from 3 to rows do A[k,2] : = `|`; od;\n print(A);\n end proc:\n\n\n\n" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 756 "LongDiv2 := proc( P, r)\n local A, C, \+ d, i,j,k,rows,q,L2,dg;\n d := degree( P); rows := 3*d +1;\n fo r k from 0 to d do C||k := coeff(P, x, k) ; od;\n A := array( \+ [seq( [ seq(` `, j = 0..(d+2) ) ], i = 1..rows) ]);\n\n A[3,1] : = x-r;\n for k from 0 to d do A[3,d - k + 3] := C||k*x^k ; od;\n \+ for k from 0 to d do A[2,d - k + 3] := `__`; od;\n A[3,2] := \+ `|`; A[4,2] := `=`; \n\n\n q := simplify(A[3,3]/x);\n A[4,1] := A[3,1]*q;\n L2 := expand(A[4,1]); dg := degree(L2);\n A[4,3] \+ := coeff(L2, x, dg)*x^dg; \n A[4,4] := coeff(L2, x, dg-1)*x^(dg-1 );\n for k from 3 to 4 do A[5,k] := `__`; od;\n A[6,3] := A[3, 3] - A[4,3]; \n A[6,4] := A[3,4] - A[4,4]; \n A[6,5] := A[3,5 ];\n print(A);\n end proc:\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1062 "LongDiv := proc( P, r)\n local A, C, d, i,j,k ,cols,rows,q,L2,dg, Q,R;\n d := degree( P); cols := d + 3; rows := 3*d + 3; \n for k from 0 to d do C||k := coeff(P, x, k) ; \+ od;\n A := array( [seq( [ seq(` `, j = 1..cols ) ], i = 1..rows ) ]);\n\n A[3,1] := x-r;\n for k from 0 to d do A[3,d - k + 3] \+ := C||k*x^k ; od;\n for k from 0 to d do A[2,d - k + 3] := `__`; od;\n A[k,2] := `|`; \n\n for k from 1 to d do \n q := si mplify(A[3*k,2+k]/x); A[1,2+k] := q;\n A[1+3*k,1] := A[3,1]* q; \+ A[1+3*k,2] := ` = `;\n L2 := expand(A[3,1]*q); dg := degree(L 2);\n A[1+3*k,2+k] := coeff(L2, x, dg) *x^dg; \n A[1+3*k ,3+k] := coeff(L2, x, dg-1) *x^(dg-1);\n if (k > 1) then A[3*k,3+ k] := A[3,3+k]; fi;\n for j from k+2 to k+3 do A[2+3*k,j] := ` __`; od;\n A[3 + 3*k,2+k] := A[3*k,2+k] - A[1+3*k,2+k]; \n \+ A[3 + 3*k,3+k] := A[3*k,3+k] - A[1+3*k,3+k]; \n od;\n print(A); \n Q := quo( P , (x-r), x, 'R'):\n print(` `);print(` `);\n \+ print(P/(x-r) = Q + R/(x-r));print(` `);\n end proc:\n\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 34 " 1. Long Division of Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 127 "\nYou are no doubt familiar with the method of convertin g an improper fraction to a mixed number by performing a long division .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "17/3; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "MixedNumber(17,3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "MixedNumber( 40,13);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "MixedNumber( 177,19 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "MixedNumber( 4001, 99 );" }}}{PARA 0 "" 0 "" {TEXT -1 170 "\nJust as we divide number using long division , we can also divide polynomials.\n\nHere are examples of this kind of division - without showing the steps of the method yet.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "PolyDivide( x^2 + 1, x - 1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "PolyDivide(x^3 + 4*x + 17 ,x -3 );\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "PolyDivide( 3*x^ 5-x^4-29*x^3-27*x^2-114*x-72, x - 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "PolyDivide( x^7 - 1, x - 3);" }}}{PARA 0 "" 0 "" {TEXT -1 125 "\nJust as with numbers, sometimes we get lucky, and the \+ denomenator goes into the numerator exactly and their is no remainder. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "PolyDivide( x^5 + 1, x + 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "PolyDivide( 3*x^5-x ^4-29*x^3-27*x^2-114*x-72, x + 3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "PolyDivide( 3*x^5-x^4-29*x^3-27*x^2-114*x-72, x - 4); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "PolyDivide( 256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 \n + 108864* x^3 + 81648*x^2 + 34992*x + 6561, \n 2*x +3 );" }}}{PARA 0 "" 0 "" {TEXT -1 80 "\nNext we'll look at the actual method to compu te these quotients and remainders." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 " 2. The Method of Long Division" }}{PARA 0 "" 0 "" {TEXT -1 98 "\nThe method of long dividing polynomials has the same four steps \+ as does numerical division :\n " }{TEXT 259 67 " 1. DIVIDE\n \+ 2. MULTIPLY\n 3. SUBTRACT\n 4. BRING DOWN" }{TEXT -1 148 "\n \nThese four steps (DMSB = \"Door Mouse Stores Bread\" ) are performed repeatedly until there is nothing more to bring down. Lets look at an example.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "(x^2 + 5*x + \+ 7) /(x+3);" }}}{PARA 0 "" 0 "" {TEXT -1 36 "\nWe first set up a long d ivision ..." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LongDiv1( x^2 + 5*x + 7, -3 );" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\nNow we go through the DMSB steps ....\n " }{TEXT 260 12 " 1. DIVIDE" }{TEXT -1 53 " ........... x into x^2 to get x" }{TEXT 261 30 " \n 2. MULTIPLY " }{TEXT -1 55 "........... x by th e divisor, x + 3 to get x^2 + 3x" }{TEXT 262 29 "\n 3. SUBTRACT \+ " }{TEXT -1 65 "........... the new product x^2 + 3x from x^2 + 5x to get 2x\n" }{TEXT 263 19 " 4. BRING DOWN" }{TEXT -1 40 " ........... 7 from the top row\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LongDiv2( x^2 + 5*x + 7, -3 );" }}}{PARA 0 "" 0 "" {TEXT -1 45 "\n\n\nNow we go through the DMSB steps AGAIN\n " } {TEXT 264 12 " 1. DIVIDE" }{TEXT -1 52 " .......... . x into 2x to get 2" }{TEXT 265 30 "\n 2. MULTIPLY \+ " }{TEXT -1 54 "........... 2 by the divisor, x + 3 to get 2x + \+ 6\n" }{TEXT 266 28 " 3. SUBTRACT " }{TEXT -1 75 "...... ..... the new product 2x + 6 from 2x + 7 on the top row to get 1\n " }{TEXT 267 19 " 4. BRING DOWN" }{TEXT -1 87 " ........... \+ (nothing more to bring down ... this means we are finished.....)\n \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "LongDiv( x^2 + 5*x + 7 , -3);" }}}{PARA 0 "" 0 "" {TEXT -1 86 "\n\nHere are other examples. T ry them by hand and verify that you get the same results. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "LongDiv( x^2 + 4*x + 10, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "LongDiv( x^2 + 1*x + 2, -2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "LongDiv( x^3 + 6*x^2 + 8*x + 2, -3);" }}} {PARA 0 "" 0 "" {TEXT -1 45 "\n\nYou can also have fractions in the di visor." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "LongDiv( 6*x^2 + 8 *x + 2, -1/3);" }}}{PARA 0 "" 0 "" {TEXT -1 140 "\nWhenever there are \+ missing terms, its necessary to put a \"placeholder\" - a zero term - \+ in its place, so that everything works smoothly. \n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "LongDiv( x^2 + 100, -5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "LongDiv( x^3 + 5*x + 7, -2);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "LongDiv( x^4 + 9*x^4 + 7, -1 );" }}}{PARA 0 "" 0 "" {TEXT -1 196 "\nThe most common places that peo ple make mistakes are when some of the numbers are negative. We must r emember to always subtract no matter whether the numbers are positive \+ or negative or mixed. \n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "LongDiv( x^2 - 5*x -2, 4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "LongDiv( x^2 + 3*x -2, 7);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 " 3. Dividing Larger Polynomials " }}{PARA 0 "" 0 "" {TEXT -1 195 "\nAbove,we divided polynomials by po lynomials of degree 1 with leading coefficient 1. We can also divide b y larger polynomials using the same method. You can do these by hand, \+ and check in Maple.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Pol yDivide( x^3 + 11*x^2 - 4*x + 17, 3*x +5);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "PolyDivide( 2000*x^3 + 5, 10*x - 9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "PolyDivide( x^4 - 1, x^2 + 1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "PolyDivide( x^4 + 1, x^2 + 1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "PolyDivide( x^4 + 2*x ^3 + 4*x^2 + 6*x + 8, x^2 + 2*x + 4);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 " 4. Factoring Polynomials " }}{PARA 0 "" 0 "" {TEXT -1 142 "\nWhen you divide polynomials and th ere is no remainder, this means that one polynomial divides the other. Thus the numerator can be factored.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "PolyDivide( 12*x^2-155*x-375 , x - 15);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "12*x^2-155*x-375 = (x - 15)*(12*x + 25);" }}}{PARA 0 "" 0 "" {TEXT -1 46 "\nWrite out the factorization f or each example." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "PolyDivi de( 15*x^3+53*x^2-69*x+400, 3*x + 16);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "PolyDivide( x^4 + 2*x^3 + 4*x^2 + 6*x + 8, x^2 + 2*x \+ + 4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PolyDivide( 168*x ^3+731*x^2-862*x+96, 7*x - 6 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "PolyDivide( 6*x^4-19*x^3-106*x^2+21*x+18, x+3 );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "PolyDivide( 6*x^4-19*x^3- 106*x^2+21*x+18, 3*x+1 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 268 35 "\n \251 2002 Waterloo M aple Inc" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }