Download 9.3 Find Special Products of Polynomials

&IND 3PECIAL 0RODUCTS OF
0OLYNOMIALS
'OAL
9OUR .OTES
+ 5SE SPECIAL PRODUCT PATTERNS TO MULTIPLY
POLYNOMIALS
315!2% /& ! ")./-)!, 0!44%2.
!LGEBRA
A B A Ê ÊÓ>LÊ B
A B A Ê ÊÓ>LÊ B
%XAMPLE
X X Ê ÊnÝÊ X X Ê Ê£ÓÝÊ %XAMPLE 5SE THE SQUARE OF A BINOMIAL PATTERN
&IND THE PRODUCT
3OLUTION
7HEN YOU USE
SPECIAL PRODUCT
PATTERNS REMEMBER
THAT A AND B CAN BE
NUMBERS VARIABLES
OR VARIABLE
EXPRESSIONS
A X X Ê ÊÓ­{Ý®­Î®Ê X Ê ÊÓ{ÝÊ B X Y X Ê ÊÓ­ÎÝ®­xÞ®Ê Y
X Ê ÊÎäÝÞÊ Y
#HECKPOINT &IND THE PRODUCT
X Ê
Ê Ý ÓÊÊ£nÝÊÊn£
X Ê
Ê {Ý ÓÊÊÓnÝÊÊ{™
R S
Ê
Ê ÓxÀ ÓÊÊ£äÀÃÊÊà Ó
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
35- !.$ $)&&%2%.#% 0!44%2.
!LGEBRA
A BA B Ê >Ê Ê LÊ %XAMPLE
X X Ê Ý Ê Ê £ÈÊ
%XAMPLE 5SE THE SUM AND DIFFERENCE PATTERN
&IND THE PRODUCT
3OLUTION
A N N Ê ˜ Ê Ê ÎÊ 3UM AND
DIFFERENCE PATTERN
Ê ˜ Ê Ê ™Ê
3IMPLIFY
B X YX Y Ê ­{Ý®Ê Ê ÞÊ Ê £ÈÝÊ Ê ÞÊ %XAMPLE 3UM AND
DIFFERENCE PATTERN
3IMPLIFY
5SE SPECIAL PRODUCTS AND MENTAL MATH
5SE SPECIAL PRODUCTS TO FIND THE PRODUCT + 3OLUTION
.OTICE THAT IS LESS THAN Ê ÓäÊ WHILE IS MORE
THAN Ê ÓäÊ + Ê ÓäÊ Ê ÓäÊ ,ESSON s !LGEBRA .OTETAKING 'UIDE
7RITE AS PRODUCT
Ê ÓäÓÊÊÎÓÊ
3UM AND DIFFERENCE
PATTERN
Ê {ääÊʙÊ
%VALUATE POWERS
Ê Î™£Ê
3IMPLIFY
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISES
&IND THE PRODUCT Z Z Ê
Ê â ÓÊÊÎÈ
&IND THE PRODUCT X X Ê
Ê £ÈÝ ÓÊʙ
&IND THE PRODUCT X YX Y
Ê
Ê Ý ÓÊÊÓxÞ Ó
$ESCRIBE HOW YOU CAN USE SPECIAL PRODUCTS TO FIND
Ê
Ê 1ÃiÊ̅iÊõÕ>ÀiʜvÊ>ÊLˆ˜œ“ˆ>Ê«>ÌÌiÀ˜Ê̜Êvˆ˜`Ê̅iÊ
«Àœ`ÕVÌÊ­{äÊÊ£®Ó°
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%XAMPLE 3OLVE A MULTISTEP PROBLEM
%YE #OLOR !N OFFSPRINGgS EYE COLOR IS DETERMINED BY A
COMBINATION OF TWO GENES ONE INHERITED FROM EACH
PARENT %ACH PARENT HAS TWO COLOR GENES AND THE
OFFSPRING HAS AN EQUAL CHANCE OF INHERITING EITHER ONE
4HE GENE " IS FOR BROWN EYES
AND THE GENE B IS FOR BLUE EYES
!NY GENE COMBINATION WITH A "
RESULTS IN BROWN EYES 3UPPOSE
EACH PARENT HAS THE SAME GENE
COMBINATION "B 4HE 0UNNETT
SQUARE SHOWS THE POSSIBLE GENE
COMBINATIONS OF THE OFFSPRING AND THE RESULTING EYE COLOR
s 7HAT PERCENT OF THE POSSIBLE GENE COMBINATIONS OF THE
OFFSPRING RESULT IN BLUE EYES
s 3HOW HOW YOU COULD USE A POLYNOMIAL TO MODEL THE
POSSIBLE GENE COMBINATIONS OF THE OFFSPRING
3OLUTION
3TEP .OTICE THAT THE 0UNNETT SQUARE SHOWS THAT Ê £Ê OUT
OF OR Ê Óx¯Ê OF THE POSSIBLE GENE COMBINATIONS
RESULT IN BLUE EYES
3TEP -ODEL THE GENE FROM EACH PARENT WITH
Ê ä°xÊÊä°xLÊ 4HE POSSIBLE GENE OF THE OFFSPRING
CAN BE MODELED BY Ê ­ä°xÊÊä°xL®ÓÊ .OTICE THAT
THIS PRODUCT ALSO REPRESENTS THE AREA OF THE
0UNNETT SQUARE
Ê ­ä°xÊÊä°xL®ÓÊ
Ê ­ä°x®ÓÊÊÓ­ä°x®­ä°xL®ÊÊ­ä°xL®ÓÊ
Ê ä°Óx ÓÊÊä°xLÊÊä°ÓxL ÓÊ
4HE COEFFICIENTS SHOW THAT Ê Óx¯Ê OF THE POSSIBLE
GENE COMBINATIONS WILL RESULT IN BLUE EYES
(OMEWORK
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISE
%YE #OLOR ,OOK BACK AT %XAMPLE 7HAT PERCENT
OF THE POSSIBLE GENE COMBINATIONS OF THE OFFSPRING
RESULT IN BROWN EYES
Çx¯
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY