{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 0 0 0 1 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 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 261 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 260 23 " Multiplying Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 208 "Exploring the methods of multiplying two polynomial s together - FOIL method, special case multiplication formulas, a more general method for multiplying arbitrary polynomials, and higher powe rs of binomials.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 153 "[Directions : Execute the Code Resource section first. \+ Although there will be no output immediately, these definitions are us ed later in this worksheet.]" }{TEXT -1 1 "\n" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "re start; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1048 "M ultPoly := proc( poly1, poly2 )\n # a = quanity of a, ar = a's r ate, and so forth for b\n local p1, p2, n1, n2,A, B,R, C,i,j,k \+ ;\n \n p1 := sort( poly1, x); p2 := sort( poly2, x);\n \+ n1 := nops(p1); n2 := nops(p2);\n\n R := array([ seq( \+ op(j-1, p1), j = 1..n1+1) ]);\n R[1] := 1; \n C := array([ seq ( op(j-1, p2), j = 1..n2+1) ]);\n C[1] := 1;\n\n A := array( [ seq( [ seq(C[k]*R[j] , j = 1..n1+1) ],\n \+ k = 1..n2+1)]);\n A[1,1] := ` `; \n B := array( \+ [seq( [ seq( ` ` , j = 1..(n1+2) ) ],\n \+ k = 1..(n2+2) )]);\n for j from 1 to n1 do B[1, j+2] := A[1,j +1]; od;\n for i from 1 to n2 do B[i+2, 1] := A[i+1,1]; od;\n\n \+ for j from 1 to n1 do B[2, j+2] := `__`; od;\n for i from 1 to n 2 do B[i+2, 2] := `|`; od;\n\n for i from 1 to n2 do \n for \+ j from 1 to n1 do \n B[i+2,j+2] := A[i+1,j+1]; \n od;od; \n print(B); print(` `); \n print(p1*p2 = sort( expand( poly1*p oly2), x )); \n end proc:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1012 "FOIL := proc( poly1, poly2 )\n \n local p1, p2, n1, n2,A, B,R, C,i,j,k, term ;\n \n p1 := sort( poly1, x); p2 := sort( poly2, x);\n n1 := nops(p1); n2 := nops(p2); \n if( (n1 = 2 ) and (n2 = 2) ) \n then\n term := s ort( expand( op(1,p1)*op(1,p2), x )); \n print(cat(`F (1s t x 1st) `,(op(1,p1)),`*`,op(1,p2),` = `),term);\n\n \+ term := sort( expand( op(1,p1)*op(2,p2), x )); \n print(ca t(`O (outer pair) `,(op(1,p1)),`*`,op(2,p2),` = `),term);\n\n \+ term := sort( expand( op(2,p1)*op(1,p2), x )); \n pri nt(cat(`I (inner pair) `,(op(2,p1)),`*`,op(1,p2),` = `),term);\n \n term := sort( expand( op(2,p1)*op(2,p2), x )); \n \+ print(cat(`L (last x last) `,(op(2,p1)),`*`,op(2,p2),` = `), term);\n\n print(` `);\n print(p1*p2 = sort( expand( poly1*poly2), x )); \n\n else\n print(`Sorry,..., FOI L only works for a product of binomails`);\n fi; \n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 38 "1. The Product of Two Binomials - Foil" }}{PARA 0 "" 0 "" {TEXT -1 360 "To multiply two binomials, we n eed to use the \"FOIL Method\". We multiply :\n \n F : fi rst x first, \n O : Outer pair (first time last) \+ \n I : Inner pair (last x first) \n L \+ : Last x Last\n\nUsually the OI terms will combine since they are l ike terms, and result will be a trinomials (three terms).\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FOIL( x + 2, x + 9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "FOIL( x + 9, x - 13);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "FOIL( x - 22, x + 18);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "FOIL( x - 36, x + 48 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "FOIL( 7*x + 16, 14*x + 24 );" }}} {PARA 0 "" 0 "" {TEXT -1 100 "\nHere is a formula version of the FOIL \+ method. However, its easier to just understand the principle." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "FOIL( a*x + b, c*x + d);" }} }{PARA 0 "" 0 "" {TEXT -1 245 "\n\nNotice that we can multiply two pol ynomials in a table format too. We put the terms of one binomial as ro w headings and the terms of the other binomial as the column headings, then complete the multiplication table, and combine the like terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "MultPoly( x + 2, 3*x + 9); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MultPoly( 100*x + 8, 12 0*x + 9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "MultPoly( 16*x + 27, 32*x + 9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "MultPo ly( .01*x + 2, .03*x + .05);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 47 "2. The Product of Two Binomials - Special Cases" }}{PARA 0 "" 0 "" {TEXT -1 136 "There are two special \+ cases which have their own formulas. We usually use these formula INST EAD of the FOIL method.\n\nFirst there is the " }{TEXT 256 21 "Differe nce of Squares" }{TEXT -1 5 ", or " }{TEXT 257 26 "Product of Conjugat e Terms" }{TEXT -1 129 ". This occurs when we multiply a binomial time s another binomial which is almost the same except the interior sign i s opposite. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "FOIL( x + \+ 2, x - 2 );" }}}{PARA 0 "" 0 "" {TEXT -1 181 "\nWe see that the O and \+ I are equal but opposite. When we combine like terms, they would cance l out. The final answer only has two terms, instead of three. Lets try another example.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "FOIL( 11*x + 13 , 11*x - 13);" }}}{PARA 0 "" 0 "" {TEXT -1 56 "Aha! It happ ened again. Lets see if there is a formula.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "FOIL( x- b, x + b);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "FOIL( a*x - b, a*x + b );" }}}{PARA 0 "" 0 "" {TEXT -1 189 "\n\nSo we can refer to this formula for doing this kind \+ of problem. First term squared minus last terms squared. Its just that simple, but it only works if the two terms are exact conjugates." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(a*x - b)*(a*x + b): % = exp and(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(3*x - 5)*(3*x + 5): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "(80* x - 66)*(80*x + 66): % = expand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 31 " \n\n\nAnother special case is the " }{TEXT 258 14 "Perfect Square" } {TEXT 259 14 " of a Binomial" }{TEXT -1 44 ". This is when a single bi nomial is squared." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "FOIL( \+ x + 5, x + 5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "MultPoly( 19*x - 3, 19*x - 3);" }}}{PARA 0 "" 0 "" {TEXT -1 146 "\nNote that th e O (outer pair) and I (inner pair) terms are equal. When we combine l ike terms, that term gets doubled. Lets get a formula for this." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "MultPoly( a*x + b, a*x + b); " }}}{PARA 0 "" 0 "" {TEXT -1 91 "\nIt appears that you square the fir st and last terms, then double the product of the terms." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "MultPoly( 3*x + 4, 3*x + 4);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MultPoly( 100*x - 8, 100*x - 8);" }}}{PARA 0 "" 0 "" {TEXT -1 210 "\n\nNote the difference between a negative constant term and a positive constant. The only difference in the final result is that the sign of the middle term is the same a s the sign of the binomial constant term." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "( 2*x - 5)*(2*x - 5): % = expand(%);\n( 2*x + 5)*(2*x + 5): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "( \+ 37*x - 17)*(37*x - 17): % = expand(%);\n( 37*x + 17)*(37*x + 17): % = \+ expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "( 1072*x - 23 2)*(1072*x - 232): % = expand(%);\n( 1072*x + 232)*(1072*x + 232): % = expand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 43 "3. The Product of Two Arbitrary Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 344 "\nWhile the FOIL method only works on two bino mails, we can still form a table to multiply two polynomials. Just as \+ before, we write each term of the one polynomial at the left and the o ther terms on top. Each term has its own row or column. We simply mult iply all the terms like a multiplication table and combine like terms \+ to get the answer.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Mult Poly( 6*x + 5, 2*x + 11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "MultPoly( x^2 + 7*x + 2, 3*x - 8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "MultPoly( x + 2, x^2 + 3*x + 9);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "MultPoly( x^2 +7*x - 13, 5*x^2 + 6*x + 4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "MultPoly( x^3 -10*x^2 + 1 1*x - 9, 10*x^2 + 17*x + 11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "MultPoly( x^3 - 13*x^2 + 6*x + 4, x^3 -8*x^2 - 6*x + 3);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "MultPoly( x^4 + 2*x^3 - 4*x^ 2 + 8*x - 16, x - 1);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\n\nHere is an \+ interesting case." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "MultPol y( x^4 + x^3 + x^2 + x + 1, x - 1);" }}}{PARA 0 "" 0 "" {TEXT -1 17 " \nHere is another." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "MultPo ly( x^4 - x^3 + x^2 - x + 1, x + 1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "4. Higher Powers of Binomial s" }}{PARA 0 "" 0 "" {TEXT -1 135 "\nWe can also find higher powers of binomials. If we square a binomial, then multiply its output by the b inomial again, we get the cube." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := 2*x + 3;\nMultPoly( A, A);\nMultPoly( A, A^2);\nA^3: % = \+ expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := 8*x - 7; \nMultPoly( A, A);\nMultPoly( A, A^2);\nA^3: % = expand(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A := 44*x + 63;\nMultPoly( A , A);\nMultPoly( A, A^2);\nA^3: % = expand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 50 "\n\nWe can also compute 4th powers in a similar way." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "A := x + 5;\nMultPoly( A, A) ;\nMultPoly( A^2, A^2);\nA^4: % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "A := 3*x + 7;\nMultPoly( A, A);\nMultPoly( A^2, \+ A^2);\nA^4: % = expand(%);" }}}{PARA 0 "" 0 "" {TEXT -1 63 "\nWe can a lso compute higher powers in the same way. For example" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "A := x + 3;\nMultPoly( A, A);\nMult Poly( A^2, A^2);\nMultPoly( A^4, A);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "A := x - 1;\nMultPoly( A, A);\nMultPoly( A^2, A^2);\n MultPoly( A^4, A^4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 263 36 "\n \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "3 0" 65 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }