5.4 Factor and Solve polynomial Equations

&ACTOR AND 3OLVE 0OLYNOMIAL
%QUATIONS
'OAL
9OUR .OTES
+ &ACTOR AND SOLVE OTHER POLYNOMIAL EQUATIONS
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0RIME POLYNOMIAL Ê«œÞ˜œ“ˆ>Ê܈̅ÊÌܜʜÀʓœÀiÊ
ÌiÀ“ÃÊ̅>ÌÊV>˜˜œÌÊLiÊÜÀˆÌÌi˜Ê>ÃÊ>Ê«Àœ`ÕVÌʜvÊ
«œÞ˜œ“ˆ>ÃʜvʏiÃÃiÀÊ`i}ÀiiÊÕȘ}ʜ˜Þʈ˜Ìi}iÀÊ
VœivvˆVˆi˜ÌÃÊ>˜`ÊVœ˜ÃÌ>˜ÌÃÊ>˜`Ê̅iʜ˜ÞÊVœ““œ˜Ê
v>V̜ÀÃʜvʈÌÃÊÌiÀ“ÃÊ>ÀiÊ£Ê>˜`Ê£
&ACTORED COMPLETELY ÊÊ«œÞ˜œ“ˆ>ÊˆÃÊv>V̜Ài`Ê
Vœ“«iÌiÞʈvʈÌʈÃÊÜÀˆÌÌi˜Ê>ÃÊ>ʓœ˜œ“ˆ>ÊœÀÊ̅iÊ
«Àœ`ÕVÌʜvÊ>ʓœ˜œ“ˆ>Ê>˜`ʜ˜iʜÀʓœÀiÊ«Àˆ“iÊ
«œÞ˜œ“ˆ>Ã°
&ACTOR BY GROUPING ʓi̅œ`ÊÕÃi`Ê̜Êv>V̜ÀÊܓiÊ
«œÞ˜œ“ˆ>ÃÊ܈̅ʫ>ˆÀÃʜvÊÌiÀ“ÃÊ̅>Ìʅ>ÛiÊ>Ê
Vœ““œ˜Ê“œ˜œ“ˆ>Êv>V̜À
1UADRATIC FORM ˜ÊiÝ«ÀiÃȜ˜ÊœvÊ̅iÊvœÀ“ÊÊ
>ÕÓÊÊLÕÊzV]Ê܅iÀiÊÕʈÃÊ>˜ÞÊiÝ«ÀiÃȜ˜Êˆ˜ÊÝ
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$EFINITION ! POLYNOMIAL WITH TWO OR MORE TERMS IS A
PRIME POLYNOMIAL IF IT Ê V>˜˜œÌÊ ÊBE WRITTEN AS A PRODUCT
OF POLYNOMIALS OF LESSER DEGREE USING ONLY INTEGER
COEFFICIENTS AND CONSTANTS AND IF THE ONLY COMMON
FACTORS OF ITS TERMS ARE Ê £Ê AND Ê £Ê %XAMPLE X X Ê ˆÃʘœÌÊ A PRIME POLYNOMIAL
BECAUSE Ê {Ê IS A COMMON FACTOR OF ALL ITS TERMS
$EFINITION ! POLYNOMIAL IS FACTORED COMPLETELY IF IT IS
WRITTEN AS A MONOMIAL OR THE PRODUCT OF A MONOMIAL
AND ONE OR MORE Ê «Àˆ“iÊ ÊPOLYNOMIALS
%XAMPLE X X zX IS NOT FACTORED COMPLETELY
BECAUSE X X Ê ­ÝzzÓ®­ÝzzÎ®Ê #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
30%#)!, &!#4/2).' 0!44%2.3
3UM OF 4WO #UBES
AzzB AzzBAzzAB B
%XAMPLE
X X Ê Ý ÓÊÊÓÝÊÊ{Ê $IFFERENCE OF 4WO #UBES
A B A BA AB B
%XAMPLE
X X Ê {Ý ÓÊÊÓÝÊÊ£Ê %XAMPLE &ACTOR THE SUM OR DIFFERENCE OF TWO CUBES
&ACTOR THE POLYNOMIAL COMPLETELY
$IFFERENCE OF
TWO CUBES
A Z Z Ê xÎÊ
Z Ê xÊ Ê â ÓÊÊxâÊÊÓxÊ &ACTOR COMMON
MONOMIAL
B Y Y Y Ê ÓÇÞ ÎÊÊÈ{Ê Y;Ê ­ÎÞ® ÎÊ Ê { ÎÊ =
3UM OF TWO
CUBES
YÊ ÎÞÊÊ{Ê Ê ™Þ ÓÊÊ£ÓÞÊÊ£ÈÊ #HECKPOINT &ACTOR THE POLYNOMIAL COMPLETELY
X n­ÝÊÊÓ®Ý ÓÊÊÓÝÊÊ{
%XAMPLE &ACTOR BY GROUPING
&ACTOR THE POLYNOMIAL X X X COMPLETELY
X X X z z X Ê ÝÊÊÓÊ Ê ÝÊÊÓÊ &ACTOR BY GROUPING
z z ­Ý ÓÊʙ®­ÝÊÊÓ®Ê
$ISTRIBUTIVE PROPERTY
z z Ê ­ÝÊÊή­ÝÊÊή­ÝÊÊÓ®Ê
$IFFERENCE OF TWO
SQUARES
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
%XAMPLE &ACTOR POLYNOMIALS IN QUADRATIC FORM
&ACTOR COMPLETELY A X AND
B Y Y Y A X Ê {Ý ÓÊ Ê £ÈÊ Ê {Ý ÓÊÊ£È{Ý ÓÊÊ£ÈÊ
Ê {Ý ÓÊʣȭÓÝÊÊ{­ÓÝÊÊ{Ê
B Y Y Y Y Ê Þ {ÊÊxÞ ÓÊÊÈÊ Ê ÎÞ ÎÞ ÓÊÊÎÞ ÓÊÊÓÊ
#HECKPOINT &ACTOR EACH POLYNOMIAL COMPLETELY
X X X ­ÝÊÊx®­ÝÊÊx®­ÝÊÊÓ®Ê
%XAMPLE X X Ê Ý ÓÊÊx­ÝÊÊή­ÝÊÊή
3OLVE A POLYNOMIAL EQUATION
7HAT ARE THE REALNUMBER SOLUTIONS OF THE EQUATION
X X X X 7RITE ORIGINAL
EQUATION
Ê Ý {ÊÊ£äÝ ÓÊÊ™Ê 7RITE IN STANDARD
FORM
Ê Ý ÓÊÊ™Ý ÓÊÊ£Ê &ACTOR TRINOMIAL
Ê ­ÝÊÊή­ÝÊÊή­ÝÊÊ£®­ÝÊÊ£®Ê $IFFERENCE OF TWO
SQUARES
X zÊ ÎÊ X Ê ÎÊ X zÊ £Ê X Ê £Ê
:ERO PRODUCT
PROPERTY
4HE SOLUTIONS ARE Ê Î]ÊÎ]z£]Ê>˜`Ê£Ê (OMEWORK
#HECKPOINT &IND THE REALNUMBER SOLUTIONS
X X X
]
]
ä]Êzq ÎÊz]ÊÊÊq ÎÊÊz]ÊÓ]ÊÓ
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE