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{SECT 0 {PARA 0 "" 0 "" {TEXT 257 49 "High School Modules > Algebra by
 Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 "    " }{TEXT 256 21 "
Factoring Polynomials" }}{PARA 0 "" 0 "" {TEXT -1 350 "\nFactoring pol
ynomials can be one of the more difficult topics in algebra. There are
 various methods, and ultimately people internalize a method based on \+
their understanding of how it works. In this worksheet, we will see ho
w to factor polynomails in Maple, and then explore a chart method of e
xamining all of the possible middle term coefficients. \n" }}{PARA 0 "
" 0 "" {TEXT 258 153 "[Directions : Execute the Code Resource section \+
first. Although there will be no output immediately, these definitions
 are used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 1161 "FactorTable  :=  proc( poly )\n local    p, lt,mt,ct, L, M, l,
m, A,i,j,k,factorable;\n     \n p := sort( poly, x); \n lt := coeff( p
, x^2);  mt := coeff( p, x); \n ct:= subs(x = 0, poly);\n print( `Lead
ing term constant` = lt);\n print( `Constant term` = ct);\n print(` `)
;\n L := numtheory[divisors](lt);  l := nops(L);  \n M := numtheory[di
visors](ct);  m := nops(M);\n \n if( ct > 0) then l := ceil(l/2); fi;
\n\n       A := array([seq([seq(` ` , j = 1..m+3) ],i = 1..l+3)]):  \n
       for i from 1 to l do \n            A[i+3, 1] := L[i];  \n      \+
      A[i+3, 2] := lt/L[i];  \n            A[i+3, 3] := `|`; od: \n   \+
    for j from 1 to m do \n            A[1, j+3] := M[j]; \n          \+
  A[2,j+3] := ct/M[j];  od:\n       for j from 1 to m do \n           \+
 A[3, j+3] := `__`; od:\n       for i from 1 to l do \n         for j \+
from 1 to m do \n            A[i+3, j+3] := A[1,j+3]*A[i+3,1] + A[2,j+
3]*A[i+3,2] ; \n         od:od:\n \n print(A); \n\nfactorable := false
;\nfor i from 1 to l do \n    for j from 1 to m do \n          if(mt =
 A[i+3, j+3]) then factorable := true ; fi;\n    od:od:\nprint(` `); \+
\nif(factorable) then print(`Factorable!`); else print(`Unfactorable`)
; fi;\n\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 595 "F
actorGrouping := proc( EQ )\n    local F, B, G, G1, G2;\n\nF := op(1, \+
EQ) + op(2, EQ): \nB := op(3, EQ) + op(4, EQ):\nprint(`\\n1. Group the
 first two terms, and last two terms :`);\nprint( EQ, `= (`,F,`) + (`,
B,`)`);  print(` `);\n\n \nG := gcd(factor(F), factor(B));\nprint(`\\n
2. These two groups are each divisible by `, G); \nG1 := simplify(F/G)
; G2 := simplify(B/G);\nprint(EQ,cat(  ` = `,G,`*`,G1,` +  `,G,`*`,G2)
);\n \n\nprint(` `);\nprint(`\\n3. Factor this common term of  `, G, `
 out`); \nG1 := simplify(F/G); G2 := simplify(B/G);\nprint(EQ,cat(  ` \+
= (`,convert(G,string) ,`)*(`,G1+G2,`)`));\n\n\nend proc:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}}{SECT 0 {PARA 4 "" 0 "
" {TEXT -1 33 "1. Factoring Polynomials in Maple" }}{PARA 0 "" 0 "" 
{TEXT -1 68 "\nMaple is just as capable of factoring polynomials as it
 is numbers." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "x^2 + 7*x + \+
12: % = factor( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "12*x
^2 + 7*x + 1: % = factor( %);" }}}{PARA 0 "" 0 "" {TEXT -1 186 "\nNote
 that NOT all polynomials are factorable. In this case, the simplest f
actorization is just the original polynomial - in the same way that th
e prime factorization of a 7 is simply 7." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "x^2 + 2*x + 3: % = factor( %);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "9*x^2 + 7*x + 11: % = factor( %);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "12*x^2 + 7*x + 1: % = factor( %);" 
}}}{PARA 0 "" 0 "" {TEXT -1 132 "\nAlso, some polynomials have factors
 common to all thier terms which can be factored out before the rest o
f the factoring continues." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "32*x^2  - 162: % = factor( %);" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\n
\nHere are some other examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "16*x^2+42*x+27: % = factor( %);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "36*x^2-105*x-50: % = factor( %);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "252*x^2+477*x+100: % = factor( %);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "252*x^2+477*x-100: % = facto
r( %);\n" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "2. Chart Method" }}
{PARA 0 "" 0 "" {TEXT -1 1891 "I created this method in the classroom.
 This method is a little more cumbersome than some of the book methods
, but it is systematic, has the same series of steps each time, and ha
s the advantage that it always gives an answer if an answer exists. An
d unlike the \"multiply the first and last terms then factor by groupi
ng\" method, this chart method leads to a good understanding of what i
s going on - which can be eventually internalized.\n\nThe OI Chart : T
he way it works is the following :\n\n        1. Find every two-number
 factorization of the leading coefficient, and the constant term\n\n  \+
      2. Create a chart where each factorization of the leading coeffi
ent has its own row along the left column,\n            and every fact
orization of the constant is in its own column along the top row.\n   \+
         (In this chart, the top two rows are the factorizations of th
e constant, and the left two rows are the\n            factorizations \+
of the leading term).\n\n       3. We are going to create the OI's (Ou
ter products and Inner products of the FOIL method). Multiply the firs
t\n            number of a l.c. factorization by the first number of a
 constant term factorization. Then multiply second \n           by sec
ond. Then add them together and put that number in the chart - at the \+
row and column which\n           align to the row and column of the tw
o factorizations.\n\n      4. When the chart is complete, look in the \+
body of the chart and see if you can find the middle term coefficient.
\n          If you can find it, then the trinomial IS factorable. If y
ou can't find it, then it's not factorable.\n\n      5. To find the fa
ctorization, trace the number in the body of the table to its row and \+
column original \n          factorizations. Form the OI computation, a
nd then form two binomials which have that Outer \n          pair prod
uct and Inner pair product. That is the factorization.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "FactorTable( x^2 + 7*x + 12); \nx^2
 + 7*x + 12: % = factor( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 66 "FactorTable( 12*x^2 + 7*x + 1); \n12*x^2 + 7*x + 1: % = factor( \+
%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "FactorTable( 9*x^2 +
 7*x + 11);\n9*x^2 + 7*x + 11: % = factor( %);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 65 "FactorTable( 12*x^2 + 7*x + 1);\n12*x^2 + 7*x \+
+ 1: % = factor( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Fac
torTable( 32*x^2  - 162);\n32*x^2  - 162: % = factor( %);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "FactorTable(16*x^2+42*x+27);\n16*x^
2+42*x+27: % = factor( %);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "FactorTable(36*x^2-105*x-50);\n36*x^2-105*x-50: % = factor( %);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "FactorTable(252*x^2+477*x+
100);\n252*x^2+477*x+100: % = factor( %);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 66 "FactorTable(252*x^2+477*x-100);\n252*x^2+477*x-100:
 % = factor( %);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "3. Factor b
y Grouping" }}{PARA 0 "" 0 "" {TEXT -1 223 " \n\"Factoring by grouping
\" is another method of factoring. One place, this method is very usef
ul is when there are four terms. The idea is to group the first two te
rms, the last two terms, and the factor out a COMMON factor." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "FactorGrouping(x^3 + 5*x^2 +
7*x + 35);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "FactorGroupin
g(x^3 - 5*x^2 -9*x + 45);\n" }}}{PARA 0 "" 0 "" {TEXT -1 129 "\n\nSome
times there is more to be done after the factoring by grouping. In thi
s case, we can still factor one of the terms further." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 30 "factor(x^3 - 5*x^2 -9*x + 45);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "FactorGrouping( 8*x^3 - 12*x
^2 - 10*x + 15);" }}}{PARA 0 "" 0 "" {TEXT -1 79 "\nThe method of fact
oring by grouping can also be used for factoring trinomials." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x^2 + 12*x + 32, ` = `, (x^2
 + 4*x),` + `,(8*x + 32);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
62 "6*x^2 + 19*x + 15;\n`6*15` = 6*15;\n`9 + 10` = 19;\n`9*10` = 90;\n
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "6*x^2 + 19*x + 15, ` =
 `, (6*x^2 + 9*x),` + `,(10*x + 15);\n6*x^2 + 19*x + 15, ` = `, 3*x,`*
(`, 2*x + 3, `) + 5*(`, 2*x + 3, `)`;\n6*x^2 + 19*x + 15, ` =  (`, 3*x
+ 5, `(`, 2*x + 3, `)`;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 33 "4. Solving Equations by Factoring" }}
{PARA 0 "" 0 "" {TEXT -1 64 "\nFactoring the polynomials in equations \+
allows us to solve them." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "
x^2 -x -12= 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "factored \+
:= factor(lhs(%)) = 0;" }}}{PARA 0 "" 0 "" {TEXT -1 202 "\nNow we use \+
the property : if a*b = 0, then a = 0 or b = 0. In other words, we sep
arate this single equation with two factors into two simpler equations
 with one factor each - which are easier to solve." }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "op(1,op(1,factored)) = 0; solve(%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "op(2,op(1,factored)) = 0; so
lve(%);" }}}{PARA 0 "" 0 "" {TEXT -1 111 "\nWe can also solve at once \+
using a single Maple command of course. But this doesn't show how we g
ot the answer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "`Solutions
 are `,\{solve( factored )\} ;" }}}{PARA 0 "" 0 "" {TEXT -1 6 "\nxxxx.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "x^2 + 39*x + 170 = 0;\n
factored := factor(lhs(%)) = 0;\nop(1,op(1,factored)) = 0; solve(%);\n
op(2,op(1,factored)) = 0; solve(%);\n`Solutions are `,\{solve( factore
d )\} ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "21*x^2 + x   - \+
10 = 0;\nfactored := factor(lhs(%)) = 0;\nop(1,op(1,factored)) = 0; so
lve(%);\nop(2,op(1,factored)) = 0; solve(%);\n`Solutions are `,\{solve
( factored )\} ;" }}}{PARA 0 "" 0 "" {TEXT -1 54 "\nOf course, this on
ly works if the polynomial factors." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "x^2  + 3*x - 11 = 0;\nfactored := factor(lhs(%)) = 0;
\n`Solutions are `,\{solve( factored )\} ;\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 61 "\nWe will use o
ther methods to solve these kinds of equations." }}}{PARA 0 "" 0 "" 
{TEXT 259 36 "\n         \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{MARK "0 0" 17 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }