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{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 34 "The Fundamental Theorem of Algebra" }}{PARA 0 "" 0 
"" {TEXT -1 67 "\nExposition and application of the fundamental theore
m of algebra.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions : Execut
e the Code Resource section first. Although there will be no output im
mediately, these definitions are used later in this worksheet.]" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " \+
" }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "c1 := 'COLOUR(RGB, .32, .32
, .64)':\nc2 := 'COLOUR(RGB, .4,  .4,  .8 )':\nc3 := 'COLOUR(RGB, .48,
 .48, .96)':\n\nc4 := 'COLOUR(RGB, .64, .16, .16 )':\nc5 := 'COLOUR(RG
B, .8,  .2,  .2 )':\nc6 := 'COLOUR(RGB, .96, .24, .24 )':" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 1 " " }{TEXT 271 33 "1. Multiple Roots & Basic Concept" }}{PARA 0 "
" 0 "" {TEXT -1 1 "\n" }{TEXT 274 13 "Example 1.1 :" }{TEXT -1 44 " \n
\nConsider this polynomial and its roots. \n" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 45 "f := x ->\n   (x+5)*(x-1)*((x-7)^2)*((x+4)^3);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( f(x) = 0, x);" }}}
{PARA 0 "" 0 "" {TEXT -1 167 "\nWhat are the distinct roots? There are
 four distinct roots : 1, 7, -4, -5. However, 7 occurs twice, and -4 i
s repeated a total of 3 times. We say that the root 7 has " }{TEXT 
272 12 "multiplicity" }{TEXT -1 18 " of 2, and -4 has " }{TEXT 273 12 
"multiplicity" }{TEXT -1 104 " 3. The multiplicity of a root is the nu
mber of times it occurs.The roots 1 and -5 have multiplicity 1.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "degree(f(x));\nexpand(f(x));
\n" }}}{PARA 0 "" 0 "" {TEXT -1 137 "\nNotice this polynomial has degr
ee 7. While f(x) has four distinct roots, it has seven roots if we cou
nt each root with its multiplicity." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "solve( f(x) = 0, x);" }}}{PARA 0 "" 0 "" {TEXT -1 
238 "\nNotice this polynomial has degree 7. While f(x) has four distin
ct roots, it has seven roots if we count each root with its multiplici
ty.\n\n\nTry to answer the following questions about each example. You
 will probably discover some rules!\n\n" }{TEXT 275 13 "Example 1.2 :
" }{TEXT -1 417 " \n\n  - What are the distinct roots of this polynomi
al?\n  - What is the multiplicity of each root?\n  - How many roots do
es it have, counting the multiplcities?\n  - What degree is the polyno
mial?\n  - What is the relationship between between the degree and the
 number of roots counting multiplicities?\n  - What is the relationshi
p between the multiplicities of a root, and the exponent on the term c
ontaining that root?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f \+
:= x ->\n   ((x-10)^3)*(x-20)*((x-30)^5)*((x-40)^2);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "degree(f(x));\nsort( expand(f(x)),x);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( f(x) = 0, x);" }}}
{PARA 0 "" 0 "" {TEXT -1 3 "\n \n" }{TEXT 276 13 "Example 1.3 :" }
{TEXT -1 417 " \n\n  - What are the distinct roots of this polynomial?
\n  - What is the multiplicity of each root?\n  - How many roots does \+
it have, counting the multiplcities?\n  - What degree is the polynomia
l?\n  - What is the relationship between between the degree and the nu
mber of roots counting multiplicities?\n  - What is the relationship b
etween the multiplicities of a root, and the exponent on the term cont
aining that root?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f := \+
x ->\n   ((x-100)^7)*((x-200)^4)*((x-300)^5)*((x-400)^3)*((x-500)^2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "degree(f(x));\nsort( ex
pand(f(x)),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "solve( f(
x) = 0, x);" }}}{PARA 0 "" 0 "" {TEXT -1 207 "\n\nCan you formulate a \+
rule about the connection between the multiplicity of a root and the e
xponent of its term?\n\nCan you formulate a rule about the degree and \+
total number of roots counting multiplicities?\n" }}}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 1 " " }{TEXT 263 37 "2. The Fundamental Theorem of Alg
ebra" }}{PARA 0 "" 0 "" {TEXT -1 5 "\n    " }{TEXT 284 4 "The " }
{TEXT 266 30 "Fundamental Theorem of Algebra" }{TEXT 286 2 " :" }
{TEXT -1 2 "\n " }{TEXT 265 3 "   " }{TEXT 268 36 "Every polynomial f(
x) has a root r.\n" }{TEXT -1 140 "\nNote that this theorem only asser
ts the existence of a root. It gives no method of finding it. In gener
al, finding roots can be difficult.\n" }{TEXT 285 1 "\n" }{TEXT 267 
13 "Example 2.1 :" }{TEXT -1 78 "  Here is a polynomial. The FTA simpl
y states that there is at least one root." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "f := x -> x^5-3*x^4+x^3+x^2-15*x+9;\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "'f(x)' = factor(f(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "plot( f(x), x = -2..3.5, color = c5);" }}}{PARA 0 "" 
0 "" {TEXT -1 1 "\n" }{TEXT 279 13 "Example 2.2 :" }{TEXT -1 77 "  The
 root, whose existence is guaranteed by the FTA may be a complex numbe
r!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f := x -> x^4+4*x^2+x+
6;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "r := (-1 + I*sqrt(7
) )/2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f(r): % = simplif
y(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "'f(x)' = (x-r)*(x
-conjugate(r))\n         *simplify(  factor(f(x))  / expand((x-r)*(x-c
onjugate(r))));" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 34 "3. Repeated Application of the
 FTA" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 282 24 "One is Never En
ough ...." }{TEXT -1 202 "\n\nIf every polynomial has a root, then it \+
can be factored into a product of a linear term and a polynomial of le
sser degree. The fundamental theorem of algebra applies to THAT smalle
r polynomial also!\n\n" }{TEXT 283 13 "Example 3.1 :" }{TEXT -1 2 " \n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "f := x -> x^4+15*x^3-70*
x^2-720*x+2304;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "'f(3)' =
 f(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "'f(x)' = '(x-3)'*
 simplify(f(x)/(x-3));" }}}{PARA 0 "" 0 "" {TEXT -1 219 "\nWe can see \+
from this example that a fourth degree polynomial is factored in a pro
duct of a first degree and third degree polynomial. We can call this t
his third degree polynomial, g(x); and, we can apply the FTA to it." }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g := 'g':\n'f(x)' = (x-3)*g
(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "g := x -> x^3+18*x^
2-16*x-768;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "'g(-8)' = g(
-8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "'g(x)' = '(x+8)'* s
implify(g(x)/(x+8));" }}}{PARA 0 "" 0 "" {TEXT -1 59 "\nWe can also ba
cktrack and express f(x) in an expanded way." }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 45 "'f(x)' = '(x-3)*(x+8)'* simplify(g(x)/(x+8));" }}}
{PARA 0 "" 0 "" {TEXT -1 93 "\n\nWe can now apply the FTA to the polyn
omial of degree two when we factor a term off of g(x)." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "h := 'h':\n'g(x)' = (x+8)*h(x);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "h := x -> x^2+10*x-96;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "'h(6)' = h(6);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "'h(x)' = '(x-6)'* simplify(h(x)/(x-
6));" }}}{PARA 0 "" 0 "" {TEXT -1 70 "\nWe can also backtrack and expr
ess f(x) in a completely factored form." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "'f(x)' = factor(f(x));" }}}{PARA 0 "" 0 "" {TEXT -1 
6 "\n\n    " }{TEXT 280 17 "Corollary to the " }{TEXT 269 30 "Fundamen
tal Theorem of Algebra" }{TEXT 281 2 " :" }{TEXT -1 6 "\n\n    " }
{TEXT 270 153 "A polynomial f(x) of degree n can be factored into prec
isely n linear terms, counting \n   multiplicities and allowing for bo
th real and complex roots.\n\n " }{TEXT -1 67 "This is a simple and us
eful result that applies to all polynomials!" }}}{SECT 0 {PARA 4 "" 0 
"" {TEXT -1 1 " " }{TEXT 264 16 "4. Complex Roots" }}{PARA 0 "" 0 "" 
{TEXT -1 343 "\nAs we saw above (in section 1), real roots might be re
peated. This is one of the ways that we we can end up with fewer disti
nct roots than the total number of roots. \n\nAnother thing that can h
appen is that some roots may be complex numbers. You might notice this
 even in this simple polynomial which has no real roots, but two compl
ex roots.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "x^2 + 1;" }}}
{PARA 0 "" 0 "" {TEXT -1 314 "\nComplex roots come in pairs. You can s
how this by taking a complex root r = a+ ib, and showing that a -ib is
 also a root. Thus the number of complex roots is always an even numbe
r. And when you multiply two linear factors with complex conjugate roo
ts, the result is a quadratic expression with real coefficients.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "(x - (a+b*I))*(x - (a-b*I)):
 % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "(x - (-5 + 3*I))*(x - (-5 - \+
3*I)): % = expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "(x
 - (71 + sqrt(32)*I))*(x - (71 - sqrt(32)*I)): % = expand(%);" }}}
{PARA 0 "" 0 "" {TEXT -1 90 "\n This means that polynomials can be fac
tored into real polynomials of degree 1 and/or 2.\n" }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 47 "5. Graphical View of Complex a
nd Repeated Roots" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 277 46 "A \+
Tale of Three Quintics :  Complex Root Pairs" }{TEXT -1 214 "\n\nLet's
 look at these three fifth degree polynomials. The first one has roots
 at 1, 2, 3, 4, and 5. The other two are related. The second one is 3 \+
greater, and the third one is 10 greater. Let's look at all three.\n" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "f := x -> (x-1)*(x-2)*(x-3
)*(x-4)*(x-5) ;\ng := x -> f(x) + 2;\nh := x -> f(x) + 6;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot( [f(x),g(x),h(x)], x = 0..6, y
 = -4..10, color = [c1,c2,c3], numpoints=1000 );" }}}{PARA 0 "" 0 "" 
{TEXT -1 166 "\nExamine the graph and count how many x intercepts each
 graph has. You will probably find that f has 5, g has 3, and h has 1 \+
root. We can verify this by solving ....\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "solve( f(x) = 0, x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "fsolve( g(x) = 0, x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "fsolve( h(x) = 0, x);" }}}{PARA 0 "" 0 "" {TEXT -1 
657 "\nWhat conclusion can we draw from this? We know that each polyno
mial, being of 5th degree, has five roots. However, the number of x-in
tercepts is 5, 3, and 1. Do multiplicities explain the lack of five di
stinct real roots? Well, no. This is clear because of the two \"hills
\" and two \"valleys\" that each graph has. The only other way we can \+
be \"missing\" the full complement of five roots, is they are complex \+
roots. We can conclude ...\n          f(x) has five distinct real root
s (of multiplicity one each)\n          g(x) has three distinct real r
oots, and one complex root pair.\n          h(x) has one distinct real
 root, and two distinct complex root pairs\n" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 90 "f := x -> (x-1)*(x-2)*(x-3)*(x-4)*(x-5) ;\ng := x \+
-> f(x) + 3;\nh := x -> expand(f(x)) + 10;" }}}{PARA 0 "" 0 "" {TEXT 
-1 77 "\nHowever, we don't need to guess or analyze. We can use Maple \+
to verify this." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fsolve( g
(x) = 0, x, complex);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "fs
olve( h(x) = 0, x, complex);" }}}{PARA 0 "" 0 "" {TEXT 278 53 "\n\n\n
\nAnother Tale of Three Quintics :  Multiplicities " }}{PARA 0 "" 0 "
" {TEXT -1 308 "\n\nLets look at these some different variations of th
at fifth degree polynomial. If we were to change some of the roots to \+
coincide with other roots, we would have multiplicities. Lets examine \+
these functions and their corresponding graphs to learn something abou
t what repeated roots look like when graphed.\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 80 "f := x -> (x-1)*(x-2)*(x-3)*(x-4)*(x-5) ;\ng :
= x -> ((x-3)^2)*(x-1)*(x-2)*(x-5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 76 "plot( [f(x),g(x)], x = 0..6, y = -10..6, color = [c2,
c6],thickness = [2,1]);" }}}{PARA 0 "" 0 "" {TEXT -1 509 "\nThe thick \+
blue graph is the original function f(x) having roots 1, 2, 3, 4, and \+
5. The thin red curve is the polynomial which has roots 1, 2, 3, and 5
 where 3 has multiplicity two. Look closely at what happens at 3. The \+
original function, f(x), passes through each of its x-intercepts. Howe
ver, at 3, g(x) \"bounces off of the x axis\" rather than pass through
. If we examine that point in greater detail, we will see that g(x) lo
oks like a mini-parabola near the point, while f(x) will look more lik
e a line.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "plot( [f(x),g
(x)], x = 2.8..3.2, y = -.1..(.1), color = [c2,c6],thickness = [2,1]);
" }}}{PARA 0 "" 0 "" {TEXT -1 116 "\n\nHere is another example, showin
g two repeated roots. The graph will show bounces at each of these rep
eated roots. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "h := x -> \+
((x-1)^2)*((x-3)^2)*(x-5);\nplot( [f(x),h(x)], x = 0..6, y = -14..6, c
olor = [c2,c5], thickness = [2,1] );" }}}{PARA 0 "" 0 "" {TEXT -1 99 "
\n\nIf we have a root of multiplicity four, and another of one, then t
here will be a similar bounce.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 104 "l := x -> ((x-2)^4)*(x-4);\nplot( [f(x),l(x)], x = 0..6, y = \+
-4..6, color = [c2,c5], thickness = [2,1] );" }}}{PARA 0 "" 0 "" 
{TEXT -1 75 "\n\nIn this case, one root has multiplicity three. How wi
ll the graph look? \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "j \+
:= x -> ((x-3)^3)*(x-1)*(x-5);\nplot( [f(x),j(x)], x = 0..6, y = -6..6
, color = [c2,c5], thickness = [2,1] );" }}}{PARA 0 "" 0 "" {TEXT -1 
319 "\nWell there is no bounce. j(x) passes through x = 3, just as f(x
) does. However, there is still something different. Can you identify \+
it? j(x) flattens out near 3, rather than passing through it as a stro
ng angle.\n\nWill it look like a mini-parabola if we blow it up? No. I
t will look more like a (negative) mini-cubic!\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 82 "plot( [f(x),j(x)], x = 2.8..3.2, y = -.01..(.0
1), color=[c2,c5],thickness=[2,1] );" }}}{PARA 0 "" 0 "" {TEXT -1 70 "
\nA similar thing happens if we make one root have multiplicity five. \+
\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "k := x -> ((x-3)^5);\n
plot( [f(x),k(x)], x = 0..6, y = -4..6, color = [c2,c5], thickness = [
2,1] );" }}}{PARA 0 "" 0 "" {TEXT -1 70 "A similar thing happens if we
 make one root have multiplicity five. \n\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 106 "plot( [f(x),g(x),h(x),j(x),k(x),l(x)], x = 0..6, y
 = -7..7, thickness = 1,\n color = [c1,c2,c3,c4,c5,c6] );" }}}{EXCHG }
}{PARA 0 "" 0 "" {TEXT 259 35 "\n         \251 2002 Waterloo Maple Inc
" }}}{MARK "0 1" 42 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }