!PPLY THE &UNDAMENTAL
4HEOREM OF !LGEBRA
'OAL
9OUR .OTES
+ #LASSIFY THE ZEROS OF POLYNOMIAL FUNCTIONS
6/#!"5,!29
2EPEATED 3OLUTION œÀÊ̅iÊiµÕ>̈œ˜Ê v ­Ý®ÊÊä]ʎÊÊ
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­ÝÊʎ®Ê…>ÃÊ>Ê`i}ÀiiÊ}Ài>ÌiÀÊ̅>˜Ê£Ê܅i˜Ê vʈÃÊ
v>V̜Ài`ÊVœ“«iÌiÞ°
4(% &5.$!-%.4!, 4(%/2%- /& !,'%"2!
4HEOREM )F F X IS A POLYNOMIAL OF DEGREE N WHERE
N z €Ê THEN THE EQUATION F X HAS AT LEAST Ê œ˜iÊ Ê
SOLUTION IN THE SET OF COMPLEX NUMBERS
#OROLLARY )F F X IS A POLYNOMIAL OF DEGREE N THEN THE
EQUATION F X HAS EXACTLY Ê ˜Ê SOLUTIONS PROVIDED
EACH SOLUTION REPEATED TWICE IS COUNTED AS Ê ÓÊ SOLUTIONS
EACH SOLUTION REPEATED THREE TIMES IS COUNTED AS Ê ÎÊ
SOLUTIONS AND SO ON
%XAMPLE &IND THE NUMBER OF SOLUTIONS OR ZEROS
&IND THE NUMBER OF SOLUTIONS OR ZEROS FOR EACH EQUATION
OR FUNCTION
A "ECAUSE X X X IS A Ê Ì…ˆÀ`Ê DEGREE
POLYNOMIAL EQUATION IT HAS Ê Ì…ÀiiÊ SOLUTIONS
B "ECAUSE F X X X X IS A Ê vœÕÀÌ…Ê DEGREE
POLYNOMIAL FUNCTION IT HAS Ê vœÕÀÊ ZEROS
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
3TATE THE NUMBER OF ZEROS OF
F X X X X ̅ÀiiÊâiÀœÃ
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
%XAMPLE &IND THE ZEROS OF A POLYNOMIAL FUNCTION
&IND ALL ZEROS OF FX X X X X X 3OLUTION
&IND THE RATIONAL ZEROS OF F "ECAUSE F IS A FIFTHDEGREE
FUNCTION IT HAS Ê vˆÛiÊ ZEROS 4HE POSSIBLE RATIONAL
ZEROS ARE Ê £]ÊÓ]ÊÎ]Ê{]ÊÈ]Ê>˜`Ê£ÓÊ 5SING
SYNTHETIC DIVISION YOU CAN DETERMINE THAT Ê ÓÊ IS A ZERO
REPEATED TWICE AND Ê £Ê IS ALSO A ZERO
7RITE FX IN FACTORED FORM $IVIDING F BY ITS KNOWN
FACTORS GIVES A QUOTIENT OF Ê Ý ÓzzÓÝzzÎÊ 3O
F X Ê ­ÝÊÊÓ®Ó­ÝÊÊ£®Ý ÓÊÊÓÝÊÊÎÊ
&IND THE COMPLEX ZEROS OF F 5SE THE QUADRATIC FORMULA
TO FACTOR THE TRINOMIAL INTO LINEAR FACTORS
]
]
F X Ê ­ÝÊÊÓ®Ó­ÝÊÊ£®Q­ÝÊÊ­£Êʈ Êq ÓÊ®zÊRQÝÊÊ­£Êʈ Êq ÓÊ®zÊRÊ
]
]
4HE ZEROS OF F ARE Ê £]ÊÓ]ÊÓ]Ê£Êʈ ÊqÓÊz]ÊÊ>˜`Ê£Êʈ ÊqÓÊzÊ
Ê Ê
#HECKPOINT &IND ALL ZEROS OF THE POLYNOMIAL FUNCTION
F X X X X X Ê £]Ê{]ÊÓÊʈ]ÊÓÊʈ
#/-0,%8 #/.*5'!4%3 4(%/2%)F F IS A POLYNOMIAL FUNCTION WITH Ê Ài>Ê COEFFICIENTS
AND Ê >ÊÊLˆÊ IS AN IMAGINARY ZERO OF F THEN Ê >ÊÊLˆÊ IS
ALSO A ZERO OF F
)22!4)/.!, #/.*5'!4%3 4(%/2%3UPPOSE F IS A POLYNOMIAL FUNCTION WITH Ê À>̈œ˜>Ê
COEFFICIENTS
AND A AND B ARE RATIONAL
NUMBERS SUCH
]
]
THAT q B]
z IS IRRATIONAL )F Ê >ÊÊÊq LÊÊz IS A ZERO OF F THEN
Ê >ÊÊÊq LÊÊz IS ALSO A ZERO OF F
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%XAMPLE 5SE ZEROS TO WRITE A POLYNOMIAL FUNCTION
7RITE A POLYNOMIAL FUNCTION F OF LEAST DEGREE THAT HAS
REAL COEFFICIENTS A LEADING COEFFICIENT OF AND AND
I AS ZEROS
"ECAUSE THE COEFFICIENTS ARE REAL AND I IS A ZERO
Ê ÎÊÊˆÊ MUST ALSO BE A ZERO 5SE THE THREE ZEROS AND THE
FACTOR THEOREM TO WRITE F X AS A PRODUCT OF THREE FACTORS
9OU CAN CHECK THIS
RESULT BY EVALUATING
F X AT EACH OF ITS
THREE ZEROS
F X Ê ÝÊÊÓÊ ;X Ê ÎÊÊˆÊ =;X Ê ÎÊÊˆÊ =
&ACTORED
FORM
Ê ÝÊÊÓÊ ;Ê ­ÝÊÊήÊÊˆÊ =;Ê ­ÝÊÊήÊÊˆÊ =
2EGROUP
TERMS
Ê ­ÝÊÊÓ®Q­ÝÊÊήÓÊʈ ÓRÊ
-ULTIPLY
Ê ­ÝÊÊÓ®QÝ ÓÊÊÈÝÊʙÊÊ­£®RÊ
%XPAND
USE
I Ê ­ÝÊÊÓ®­Ý ÓÊÊÈÝÊÊ£ä®Ê
3IMPLIFY
Ê Ý ÎÊÊÈÝ ÓÊÊ£äÝÊÊÓÝ ÓÊÊ£ÓÝÊÊÓäÊ
-ULTIPLY
Ê Ý ÎÊÊ{Ý ÓÊÊÓÝÊÊÓäÊ
#OMBINE
LIKE TERMS
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
7RITE A POLYNOMIAL OF LEAST DEGREE THAT HAS RATIONAL
COEFFICIENTS
A LEADING COEFFICIENT OF AND AND
]
q z AS ZEROS
Ê
Ê v ­Ý®ÊÊÝ ÎÊÊÈÝ ÓÊÊÎÝÊÊÓä
$%3#!24%3 25,% /& 3)'.3
,ET F X AN X N AN X N A X AX A
BE A POLYNOMIAL FUNCTION WITH REAL COEFFICIENTS
s 4HE NUMBER OF Ê «œÃˆÌˆÛiÊ REAL ZEROS OF F IS EQUAL
TO THE NUMBER OF CHANGES IN SIGN OF THE COEFFICIENTS
OF Ê v ­Ý®Ê OR IS LESS THAN THIS BY AN Ê iÛi˜Ê NUMBER
s 4HE NUMBER OF Ê ˜i}>̈ÛiÊ REAL ZEROS OF F IS EQUAL
TO THE NUMBER OF CHANGES IN SIGN OF THE COEFFICIENTS
OF Ê v ­Ý®Ê ÊOR IS LESS THAN THIS BY AN Ê iÛi˜Ê NUMBER
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
%XAMPLE 5SE $ESCARTES RULE OF SIGNS
$ETERMINE THE POSSIBLE NUMBERS OF POSITIVE REAL ZEROS
NEGATIVE REAL ZEROS AND IMAGINARY ZEROS FOR
F X X X X X X 3OLUTION
FX X X X X X 4HE COEFFICIENTS IN F X HAVE Ê ÓÊ SIGN CHANGES SO F HAS
Ê ÓʜÀÊäÊ POSITIVE REAL ZEROS
F X X X X X X F X Ê ÓÝ xÊÊÇÝ {ÊÊ£ÓÝ ÎÊÊÓÝ ÓÊÊ{ÝÊÊÈÊ
4HE COEFFICIENTS IN F X HAVE Ê ÎÊ SIGN CHANGES SO F
HAS Ê ÎʜÀÊ£Ê NEGATIVE REAL ZEROS
0OSITIVE REAL
ZEROS
.EGATIVE REAL
ZEROS
)MAGINARY
ZEROS
4OTAL
ZEROS
Ê ÓÊ
Ê ÎÊ
Ê äÊ
Ê xÊ
Ê ÓÊ
Ê £Ê
Ê ÓÊ
Ê xÊ
Ê äÊ
Ê ÎÊ
Ê ÓÊ
Ê xÊ
Ê äÊ
Ê £Ê
Ê {Ê
Ê xÊ
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
$ETERMINE THE POSSIBLE NUMBERS OF POSITIVE REAL
ZEROS NEGATIVE REAL ZEROS AND IMAGINARY ZEROS FOR
F X X X X X X (OMEWORK
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x
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ä
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x
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,ESSON s !LGEBRA .OTETAKING 'UIDE