&ACTOR 0OLYNOMIALS #OMPLETELY
'OAL
9OUR .OTES
+ &ACTOR POLYNOMIALS COMPLETELY
6/#!"5,!29
&ACTOR BY GROUPING >V̜Àˆ˜}Ê>Ê«œÞ˜œ“ˆ>Ê܈̅ÊvœÕÀÊ
ÌiÀ“ÃÊLÞÊv>V̜Àˆ˜}Ê>ÊVœ““œ˜Ê“œ˜œ“ˆ>ÊvÀœ“Ê>Ê«>ˆÀÊ
œvÊÌiÀ“Ã]Ê̅i˜ÊœœŽˆ˜}ÊvœÀÊ>ÊVœ““œ˜ÊLˆ˜œ“ˆ>ÊÌiÀ“
&ACTOR COMPLETELY >V̜Àˆ˜}Ê>Ê«œÞ˜œ“ˆ>Ê՘̈ÊˆÌʈÃÊ
ÜÀˆÌÌi˜Ê>ÃÊ>Ê«Àœ`ÕVÌʜvÊ՘v>V̜À>LiÊ«œÞ˜œ“ˆ>ÃÊ
܈̅ʈ˜Ìi}iÀÊVœivvˆVˆi˜ÌÃ
%XAMPLE &ACTOR OUT A COMMON BINOMIAL
&ACTOR THE EXPRESSION
B Y Y Y
A XX zX 3OLUTION
A XX zX X Ê ÎÝÊÊÓÊ B 4HE BINOMIALS Y AND Y ARE Ê œ««œÃˆÌiÃÊ &ACTOR
Ê £Ê FROM Y TO OBTAIN A COMMON BINOMIAL FACTOR
Y Y Y Y Y Ê Î­ÞÊÊ{®Ê
zY Ê ­Þ ÓÊÊήÊ
%XAMPLE &ACTOR BY GROUPING
&ACTOR THE EXPRESSION
A Y Y zY 2EMEMBER THAT
YOU CAN CHECK A
FACTORIZATION BY
MULTIPLYING THE
FACTORS
B Y Y YX X
3OLUTION
A Y Y zY Ê Þ ÎÊÊÇÞ ÓÊ Ê ÓÞ ÊÊ£{Ê Ê Þ ÓÊ Ê Þ ÊÊÇÊ Ê ÓÊ Ê Þ ÊÊÇÊ Ê Þ ÊÊÇÊ Ê Þ ÓÊÊÓÊ B Y Y YX X zÊ Þ ÓÊÊÓÞ Ê Ê ÞÝÊÊÓÝÊ ,ESSON s !LGEBRA .OTETAKING 'UIDE
zÊ ÞÊ Ê Þ ÊÊÓÊ Ê ÝÊ Ê Þ ÊÊÓÊ Ê Þ ÊÊÓÊ Ê Þ ÊÊÝÊ #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
%XAMPLE &ACTOR BY GROUPING
&ACTOR X X X 3OLUTION
4HE TERMS X AND HAVE NO COMMON FACTOR 5SE
THE Ê Vœ““ÕÌ>̈ÛiÊ«Àœ«iÀÌÞÊ TO REARRANGE THE TERMS SO
THAT YOU CAN GROUP TERMS WITH A COMMON FACTOR
X X X Ê Ý ÎÊÊ{Ý ÓÊÊÎÝÊÊ£ÓÊ
zÊ ­Ý ÎÊÊ{Ý Ó®ÊÊ­ÎÝÊÊ£Ó®Ê
Ê Ý Ó­ÝÊÊ{®ÊÊέÝÊÊ{®Ê
Ê ­ÝÊÊ{®­Ý ÓÊÊήÊ
#HECKPOINT &ACTOR THE EXPRESSION
ZZ Z Ê
Ê ­âÊÊÈ®­xâÊÊ{®Ê
X X X Ê
Ê ­ÝÊÊ{®­Ý ÓÊÊx®Ê
Y Y Y
Ê ­ÞÊÊ£®­ÓÞ ÓÊÊÇ®
N N N
Ê ­˜ÊÊn®­˜ ÓÊÊÈ®
'5)$%,).%3 &/2 &!#4/2).' 0/,9./-)!,3
#/-0,%4%,9
4O FACTOR A POLYNOMIAL COMPLETELY YOU SHOULD TRY EACH
OF THESE STEPS
&ACTOR OUT THE Ê }Ài>ÌiÃÌÊ COMMON MONOMIAL FACTOR
,OOK FOR A DIFFERENCE OF TWO SQUARES OR A Ê «iÀviVÌÊ
õÕ>ÀiÊÌÀˆ˜œ“ˆ>Ê &ACTOR A TRINOMIAL OF THE FORM AX BX C INTO A
PRODUCT OF Ê Lˆ˜œ“ˆ>Ê FACTORS
&ACTOR A POLYNOMIAL WITH FOUR TERMS BY Ê }ÀœÕ«ˆ˜}Ê #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
&ACTOR COMPLETELY
%XAMPLE &ACTOR THE POLYNOMIAL COMPLETELY
A X X B R R R
C D D
3OLUTION
A 4HE TERMS OF THE POLYNOMIAL HAVE NO COMMON
MONOMIAL FACTOR !LSO THERE ARE NO FACTORS OF Ê £Ê
THAT HAVE A SUM OF Ê ÎÊ 4HIS POLYNOMIAL Ê V>˜˜œÌÊ
BE FACTORED
B R R R Ê ÎÀ ­À ÓÊÊÇÀÊÊ£ä®Ê
ÎÀ ­À ÊÊÓ®­À ÊÊx®Ê
C D D Ê ` Ó­™` ÓÊÊ{®Ê
Ê ` Ó­Î` ÊÊÓ®­Î` ÊÊÓ®Ê
3OLVE A POLYNOMIAL EQUATION
%XAMPLE 3OLVE X X X
3OLUTION
X X X
7RITE ORIGINAL
EQUATION
X X Ê ÊÊ X 2EMEMBER THAT
YOU CAN CHECK
YOUR ANSWERS BY
SUBSTITUTING EACH
SOLUTION FOR X IN THE
ORIGINAL EQUATION
Ê xÝ ­Ý ÓÊÊxÝ ÊÊÈ®Ê &ACTOR
OUT Ê xÝÊ Ê xÝ ­ÝÊÊή­ÝÊÊÓ®Ê &ACTOR
TRINOMIAL
Ê xÝÊÊäÊ OR Ê ÝÊÊÎÊÊäÊ OR Ê ÝÊÊÓÊÊäÊ
X Ê äÊ
Ê ``Ê X
TO EACH SIDE
,ESSON s !LGEBRA .OTETAKING 'UIDE
X Ê ÎÊ
X Ê ÓÊ
:EROPRODUCT
PROPERTY
3OLVE FOR X
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
%XAMPLE 3OLVE A MULTISTEP PROBLEM
6OLUME ! CRATE IN THE SHAPE OF A RECTANGULAR PRISM HAS A
VOLUME OF CUBIC FEET 4HE CRATE HAS A WIDTH OF W FEET
A LENGTH OF W FEET AND A HEIGHT OF W FEET 4HE
LENGTH IS MORE THAN HALF THE WIDTH &IND THE CRATEgS LENGTH
WIDTH AND HEIGHT
3OLUTION
3TEP 7RITE AND SOLVE AN EQUATION FOR W
6OLUME Ê i˜}Ì…Ê + Ê Üˆ`Ì…Ê + Ê …iˆ}…ÌÊ
Ê £näÊ Ê ­™ÊÊÜ®­Ü®­ÜÊÊ{®Ê
Ê Ü ÎÊÊxÜ ÓÊÊÎÈÜÊÊ£näÊ
Ê Ü Ó­ÜÊÊx®ÊÊÎÈ­ÜÊÊx®Ê
Ê ­ÜÊÊx®­Ü ÓÊÊÎÈ®Ê
Ê £­ÜÊÊx®­Ü ÓÊÊÎÈ®Ê
Ê £­ÜÊÊx®­ÜÊÊÈ®­ÜÊÊÈ®Ê
Ê ÜÊÊxÊ OR Ê ÜÊÊÈÊ OR Ê ÜÊÊÈÊ W Ê xÊ
W Ê ÈÊ
W Ê ÈÊ
3TEP #HOOSE THE SOLUTION THAT IS THE CORRECT VALUE FOR
W $ISREGARD Ê ÈÊ BECAUSE THE WIDTH CANNOT
BE Ê ˜i}>̈ÛiÊ 9OU KNOW THAT THE LENGTH IS MORE THAN HALF THE
WIDTH 4EST THE SOLUTIONS Ê xÊ>˜`ÊÈÊ IN THE LENGTH
EXPRESSION
,ENGTH Ê ™ÊÊxÊ Ê {Ê OR
,ENGTH Ê ™ÊÊÈÊ Ê ÎÊ 4HE SOLUTION Ê xÊ GIVES A LENGTH OF Ê {Ê FEET WHICH
IS MORE THAN HALF THE WIDTH
3TEP &IND THE HEIGHT
(EIGHT Ê ÜÊÊ{Ê Ê xÊÊ{Ê Ê ™Ê 4HE WIDTH IS Ê xÊviiÌÊ THE LENGTH IS Ê {ÊviiÌÊ AND THE
HEIGHT IS Ê ™ÊviiÌÊ #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
#HECKPOINT &ACTOR THE POLYNOMIAL
X X X
ÓÝ­ÝÊÊÈ®­ÝÊʙ®
Ê Y Y ÎÞ Ó­ÓÞÊÊx®­ÓÞÊÊx®
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISES
3OLVE X X X
ä]Êx]Ê{
(OMEWORK
7HAT )F ! BOX IN THE SHAPE OF A RECTANGULAR PRISM
HAS A VOLUME OF CUBIC FEET 4HE BOX HAS A
LENGTH OF X FEET A WIDTH OF X FEET AND A HEIGHT
OF X FEET &IND THE DIMENSIONS OF THE BOX
ÈÊviiÌʏœ˜}ÊLÞÊ£xÊviiÌÊ܈`iÊLÞÊÓÊviiÌʅˆ}…
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY