3OLVE 0OLYNOMIAL %QUATIONS
IN &ACTORED &ORM
'OAL
9OUR .OTES
+ 3OLVE POLYNOMIAL EQUATIONS
6/#!"5,!29
2OOTS /…iÊ܏Ṏœ˜ÃʜvÊ>˜ÊiµÕ>̈œ˜Êˆ˜Ê܅ˆV…Êœ˜iÊ
È`iʈÃÊâiÀœÊ>˜`Ê̅iʜ̅iÀÊÈ`iʈÃÊ>Ê«Àœ`ÕVÌʜvÊ
«œÞ˜œ“ˆ>Êv>V̜ÀÃ
6ERTICAL MOTION MODEL ʓœ`iÊ̅>ÌÊ`iÃVÀˆLiÃÊ̅iÊ
…iˆ}…ÌʜvÊ>Ê«ÀœiV̈i
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,ET A AND B BE REAL NUMBERS )F AB THEN Ê >Ê OR Ê LÊ 5SE THE ZEROPRODUCT PROPERTY
%XAMPLE 3OLVE X X 3OLUTION
X zX z Ê ÝÊÊxÊ OR Ê ÝÊÊ{Ê X Ê xÊ OR
X Ê {Ê
7RITE ORIGINAL
EQUATION
Ê <iÀœ‡«Àœ`ÕVÌÊ Ê
PROPERTY
3OLVE FOR X
4HE SOLUTIONS OF THE EQUATION ARE Ê xÊ>˜`Ê{Ê #(%#+ 3UBSTITUTE EACH SOLUTION INTO THE ORIGINAL
EQUATION TO CHECK
Ê xÊ Ê xÊ gg
Ê {Ê Ê {Ê gg
Ê äÊ+Ê™Ê gg
Ê ™Ê+ÊäÊ gg
Ê äÊ #OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
Ê äÊ ,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%XAMPLE &IND THE GREATEST COMMON MONOMIAL FACTOR
&ACTOR OUT THE GREATEST COMMON MONOMIAL FACTOR
B X X
A X Y
3OLUTION
A 4HE '#& OF AND IS Ê nÊ 4HE VARIABLES X AND Y
HAVE Ê ˜œÊVœ““œ˜Êv>V̜ÀÊ 3O THE GREATEST COMMON
MONOMIAL FACTOR OF THE TERMS IS Ê nÊ X Y Ê n­ÓÝÊÊxÞ®Ê
B 4HE '#& OF AND IS Ê ÈÊ 4HE '#& OF X AND X IS
Ê Ý ÓÊ 3O THE GREATEST COMMON MONOMIAL FACTOR OF THE
TERMS IS Ê ÈÝ ÓÊ X X Ê ÈÝ Ó­£ÊÊxÝ®Ê
%XAMPLE 3OLVE AN EQUATION BY FACTORING
3OLVE THE EQUATION
A
X X /RIGINAL EQUATION
Ê ÎÝ­ÝÊÊx®Ê &ACTOR LEFT SIDE
Ê ÎÝÊ :EROPRODUCT PROPERTY
OR Ê ÝÊÊxÊ X Ê äÊ OR
X Ê xÊ
3OLVE FOR X
4HE SOLUTIONS OF THE EQUATION ARE Ê äÊ>˜`ÊxÊ 4O USE THE ZERO
PRODUCT PROPERTY
YOU MUST WRITE THE
EQUATION SO THAT
ONE SIDE IS &OR
THIS REASON Ê Ó{LÊ
MUST BE SUBTRACTED
FROM EACH SIDE OF
THE EQUATION
B B
B
/RIGINAL EQUATION
Ê ™LÓÊÊÓ{LÊ 3UBTRACT Ê Ó{LÊ FROM
EACH SIDE
Ê ÎL­ÎLÊÊn®Ê &ACTOR LEFT SIDE
Ê ÎLÊ :EROPRODUCT PROPERTY
OR Ê ÎLÊÊnÊ B Ê äÊ OR
BÊ]
ÊnÊz
Î
3OLVE FOR B
4HE SOLUTIONS OF THE EQUATION ARE Ê äÊ>˜`ÊÊ]
ÊnÊz Î
,ESSON s !LGEBRA .OTETAKING 'UIDE
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9OUR .OTES
#HECKPOINT 3OLVE THE EQUATION
X X Ê
Ê È]ÊÎ
X X n]Êx
#HECKPOINT &ACTOR OUT THE GREATEST COMMON
MONOMIAL FACTOR
X Y
Ê
Ê Ó­xÝ ÓÊÊ£ÓÞ Ó®
T T Ì {­ÎÌ ÓÊÊn®
4HE VERTICAL MOTION
MODEL TAKES
INTO ACCOUNT THE
EFFECT OF GRAVITY
BUT IGNORES OTHER
LESS SIGNIFICANT
FACTORS SUCH AS AIR
RESISTANCE
6%24)#!, -/4)/. -/$%,
4HE HEIGHT H IN FEET OF A PROJECTILE CAN BE MODELED BY
H T VT S
WHERE T IS THE Ê Ìˆ“iÊ IN SECONDS THE OBJECT HAS BEEN
IN THE AIR V IS THE Ê ˆ˜ˆÌˆ>ÊÛiÀ̈V>ÊÛiœVˆÌÞÊ IN FEET PER
SECOND AND S IS THE Ê ˆ˜ˆÌˆ>Ê…iˆ}…ÌÊ IN FEET
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,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
%XAMPLE 3OLVE A MULTISTEP PROBLEM
&OUNTAIN ! FOUNTAIN SPRAYS WATER INTO THE AIR WITH AN INITIAL
VERTICAL VELOCITY OF FEET PER SECOND !FTER HOW MANY
SECONDS DOES IT LAND ON THE GROUND
3OLUTION
3TEP 7RITE A MODEL FOR THE WATERgS HEIGHT ABOVE GROUND
H T VT S
6ERTICAL MOTION MODEL
H T Ê ÓäÊ T Ê äÊ
V Ê ÓäÊ AND S Ê äÊ
H T Ê ÓäÌÊ
3IMPLIFY
3TEP 3UBSTITUTE Ê äÊ FOR H 7HEN THE WATER LANDS ITS
HEIGHT ABOVE THE GROUND IS Ê äÊ FEET 3OLVE FOR T
4HE SOLUTION
T MEANS THAT
BEFORE THE WATER IS
SPRAYED ITS HEIGHT
ABOVE THE GROUND IS
FEET
Ê äÊ T Ê ÓäÌÊ
3UBSTITUTE Ê äÊ FOR H
Ê äÊ Ê {Ì­{ÌÊÊx®Ê
&ACTOR RIGHT SIDE
Ê {ÌÊÊäÊ OR Ê {ÌÊÊxÊÊäÊ
:EROPRODUCT PROPERTY
Ê ÌÊÊäÊ OR Ê ÌÊÊ£°ÓxÊ
3OLVE FOR T
4HE WATER LANDS ON THE GROUND Ê £°ÓxÊ SECONDS AFTER IT
IS SPRAYED
#HECKPOINT #OMPLETE THE FOLLOWING EXERCISES
3OLVE D D ä]ÊÇÊ
(OMEWORK
3OLVE B B
Ê ä]ÊÊ]
Ê£Êz
{
7HAT )F )N %XAMPLE SUPPOSE THE INITIAL VERTICAL
VELOCITY IS FEET PER SECOND !FTER HOW MANY
SECONDS DOES THE WATER LAND ON THE GROUND
£°£ÓxÊÃiVœ˜`Ã
,ESSON s !LGEBRA .OTETAKING 'UIDE
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