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Tharp 8How to factor a trinomial of the form ax2+bx+c when a=1B9)fnf*To factor a trinomial means to write it as a product of two binomials. For example x2+6x+8= (x+2) (x+4) two binomials Remember we can use F.O.I. L. to multiply out (x+2) (x+4) we get: ctoring trinomials is an important Algebraic skill. It is used in higher-level mathematics and science. Besides that you will need it to simplify some polynomial expressions and solve more complex equations. fZZjZFgc gog gBgcGgcgc8"*'1"',2sTo get this trinomial into its factored form we use F.O.I. L. in Reverse on x2+ 6x+8.ptR'''/'  Remember that the FIRST term x2 came from its factors x (x). So these two factors become the first terms in each binomial. r#''''/''('!''#'#M'#'""@8To do this we look at the middle terms of x2+2x+4x+8.9.'/''''''q When we use F.O. I. L. to multiply the OUTSIDE and INSIDE terms in (X+2) (x+4) r)'''''% % %+Lets make a table to organize what we know.*,+f" ?Interactive FOIL Exploration Print this page.,@ BAReview To Factor A Trinomial ax2+bx+c when a=1 Print this page.ZBBPractice Problems KNow click on the door to go on to How to factor the form ax2+bx+c with a>1TL#BbBJB!0#:  ` f [f f3̙` 999f~` ___` 33f3f` f33f 3f3̙` 33f̙3f>?" dd@(33?Zd@ 33 d " @ `"  n?" dd@   @@``PP   @ ` ` p>> H@ <(  <|T  < " <  fYgֳgֳd?P < 2  <  `p Ygֳgֳ? < 2 Z < s *A?8 < # l|Y1 ?"  Y T Click to edit Master title style! ! <  fY1 ?"` Y ,  < # lY1 ?" Y >*  < # lY1 ?"  Y @*  < # l!Y1 ?" Y @*B < s *? ? f [f f3̙ Lock And Key  sk @ (  @|T  @ " @  f(Ygֳgֳd? < 2  @  `t,Ygֳgֳ?`  < 2 Z @ s *A?|  @ # l0Y1 ?" U  Y T Click to edit Master title style! ! @  f3Y1 ?" t Y W#Click to edit Master subtitle style$ $ @ # l8Y1 ?"0  Y >*  @ # l;Y1 ?"   Y @*  @ # l>Y1 ?" Y @*B @ s *? ? f [f f3̙  0 6(   x  c $]@ U  ] ~  s *xY@ t ] H  0޽h ? f [f f3̙\    @ (    s *Lv< <$D 0  v   s *]<`<$0 v   C XA@D:\Mth03\TRINOM\TRI3\IMG001.PNG  ,$D 0%   `ȫ$331?p @,$D  0 aX2+6x+8> 233 3333B @  `D33>? ` ,$D 0B @  `D33>? P ,$D 0B    `DB>? ,$D 0B  @  `DB>? ,$D 0B    `D33>?@ P ,$D 0B    `D33>? P ,$D 0B    `DB>?  ,$D  0B   `DB>?  ,$D  0B   `DBԔ? @ ` ,$D  0B   `DBԔ? ,$D 0   f8$1? ,$D 0 : 2 K   f%1?P 0,$D  0 %Adding the middle two terms you get @%0ZZ 2%%H  0޽h ? f [f f31  P q(  ~  s *]<   ]   c $]<p  <$ 0 ]   C XA@D:\Mth03\TRINOM\TRI3\IMG005.PNG@@,$D 0  C ZA BD:\Mth03\TRINOM\TRI3A\IMG004.PNG` ,$D 0'   f$1? ,$D 0 ]]Remember that the LAST term in each binomial came from factors of the LAST term 8 in x2+ 6x+8^ 2''0' '''''%-%^    f$1?  ,$D 0 So 8 breaks down to 1(8) or 2(4). We need to choose which pair of factors 1 and 8 or 2 and 4 we want to become the last terms in each binomial 0ZZ 2''''+''''4'H  0޽h ? f [f f3̙  1)(  r  S 1$<   $   c $8<`<$0    C ZA BD:\Mth03\TRINOM\tri3b\IMG002.PNG" ` ,$D0   f0J1?  ,$D0 @The inside and outside products are 2x and 4x. Their sum is 6x. @0ZZ 2$'''' '''   f0w_1? `),$D0 So we are looking for factors of 8 (the constant term) that add to 6 to get the coefficient of the middle term in x2+ 6x+8.  Z 2% %% %,%%% %%-%%% H  0޽h ? f [f f3̙<   #D|(  Dr D S w$<   $ ^  0  2D #"  0  ,D # l1? 0  P  @`B -D T1 ? 0 0B .D T1 ? ~B /D N1 ? 0 B 0D T1 ? 0  D # l0$1?`` ,$D 0 i We use a table to find two factors of 8 who sum to 6 the coefficient of the middle term in x2+6x+8. j0ZZ2]goge%z   D W ,$D 0 @   D @ z Dz  D # l$1?+O g Factors of 8 >   '"  D  `?z@ z  Dz  D # lV`1?a  pSums of These Factors > '"  D  `?z @ z Dz D # l<`1?+O c 1 and 8 >  '"  D  `?z@ z  Dz  D # lL|_1?a  `1+8=9 > '"  D  `?z @ z Dz D Z1?z@ z Dz D  f_1?+O b2 and 4 > '"  D  `?z@ z  Dz  D Z1?z @ z  Dz  D  fH]1?a  `2+4=6 > '"  D  `?z  D Z? ] D  f$1? ,$D 0 So 2 and 4 have a sum of 6 and a product of 8. So these are the factors of 8 we use as the last terms of the binomial (x+__)(x+__). Therefore x2+6x+8= (x+2) (x+4) 0ZZ2 2ggggggggggggdgo gggggH D 0޽h ? f [f f3̙3  0Hs(  Hr H S xs<   $   H  fP1?   ,$D 0 CFirst get the Live Math Plug in at: http://www.livemath.com/download/ Be sure to download the plug in for your browser Internet Explorer or Netscape.0 Z33*$ Q!0$E H  f81?  ,$D 0 Next click on the link to the Live Math Exploration and follow the directions in it. http://www.tcc.edu/VML/Mth03/TRINOM/trinom3.htm * 2U33133!0U H  fT1?` p,$D 0 Finally click the back button on your browser to come back to this lesson. If for some reason this page does not work click back until you get to this presentation ignore the link and click past this page., 233H H 0޽h ? f [f f3x  ( PP(  Pr P S ,<   $  P  f1?`*  y} 1) Factor x2 into x (x). 2) Find two numbers who are factors of the last term (constant term), c, and whose sum equals the coefficient of the middle term (x-term) b. 3) Use the two numbers found in the above step, including their signs, to write the trinomial in factored form. The trinomial in factored form will be: (X + a number) (X + a number)  2} 2 '/ ''"'"3'" P  f1? P  Y#Now go on to the practice problems.$ 2$33H P 0޽h ? f [f f3̙$  `Td(  Tr T S L.<   $  T  fT1?0``XP___PPT92* Try this practice. Factor each trinomial below. x2+7x+12 answer 2) x2-6x+8 answer 3) x2-18x+80 answer 4) x2-2x-80 answer 5) b2-11b+18 answer 6) x2+8xy+15y2 answer 0 2 2 2% % %-%%-%-% %-% $ %-%  $,$,% r@!0AG!0_e!0{!0!0!0 @`H T 0޽h ? f [f f3̙v  &pX(  Xr X S $<   $  X c Ac:\Program Files\Microsoft Office\Clipart\standard\stddir1\bd07236_.wmfLP   !0H X 0޽h ? f [f f3̙r@ 8.'0/2s@`H5Pyip4v`} 0(   )VBhttp://www.livemath.com/download/r^http://www.tcc.edu/VML/Mth03/TRINOM/trinom3.htmT@D:\Mth03\Trinom\Solution.htm#oneVBD:\Mth03\Trin Oh+'0$R hp   @ L XdlPowerPoint PresentationDr. Marcia L. TharpRC:\Program Files\Microsoft Office\Templates\Presentation Designs\Lock And Key.potrTCC13Microsoft PowerPointoso@}9#@`y@.V)jGPg  P('& &&#TNPP2OMi & TNPP &&TNPP     'A x(xʦ """)))UUUMMMBBB999|PP3f3333f333ff3fffff3f3f̙f3333f3333333333f3333333f3f33ff3f3f3f3333f3333333f3̙33333f333ff3ffffff3f33f3ff3f3f3ffff3fffffffff3fffffff3f̙ffff3ff333f3ff33fff33f3ff̙3f3f3333f333ff3fffff̙̙3̙f̙̙̙3f̙3f3f3333f333ff3fffff3f3f̙3ffffffffff!___wwwC_=_C_=_C_=_C_=_C_=_C_=_C_=_C_=__=_`_=_`_=_`_=_`_=_`_=_`_=_`_=_`=_C`=_C`=_C`=_C`=_C`=_C`=_C`=_C`_`_`_`_`_`_`_`_`_`_`_`_`_`_`_`_`C`C`C`C`C`C`C`C`C`C`C`C`C`C`C`C``C`f`C`f`C`f`C`f`C`f`C`f`C`f`C`fC`C`C`C`C`C`C`C`C`C`C`C`C`C`C`C`afafafaCafafafaCafafafaCafafafaCCfCfCfCfCfCfCfCfCfCfCfCfCfCfCfCfgCgfgCgfgCgfgCgfgCgfgCgfgCgfgCgfCgCgCgCgCgCgCgCgCgCgCgCgCgCgCgCgmggCmggCmggCmggCmggCmggCmggCmggCCgCgCgCgCgCgCgCgCgCgCgCgCgCgCgCgmDmgmDmDmDmgmDmDmDmgmDmDmDmgmDmDDmDgDmDgDmDgDmDgDmDgDmDgDmDgDmDgmgmDmDmDmgmDmDmDmgmDmDmDmgmDmDmDEmEmEmEmEmEmEmEmEmEmEmEmEmEmEmEmmEmDmDmJmEmDmDmJmEmDmDmJmEmDmDmJEmEmEmEmEmEmEmEmEmEmEmEmEmEmEmEmmKmKmKmKmKmKmKmKmKmKmKmKmKmKmKmKEKKmEKKmEKKmEKKmEKKmEKKmEKKmEKKmmKmKmKmKmKmKmKmKmKmKmKmKmKmKmKmKKnKKKnQKKnKKKnQKKnKKKnQKKnKKKnQKnKnKnKnKnKnKnKnKnKnKnKnKnKnKnKnKQKQKQKQKQKQKQKQKQKQKQKQKQKQKQKQKKQoKnQoKKQoKnQoKKQoKnQoKKQoKnQoKQKQKQLQKQKQKQLQKQKQKQLQKQKQKQLQKoQLQoKoQoQLQoKoQoQLQoKoQoQLQoKoQQLQLQQQLQLQLQQQLQLQLQQQLQLQLQQQLLQoRLQLQLQoRLQLQLQoRLQLQLQoRLQLQQLQRQLQRQLQRQLQRQLQRQLQRQLQRQLQRoRLRoRLRoRLRoRLRoRLRoRLRoRLRoRLRQRQRQRRRQRQRQRRRQRQRQRRRQRQRQRRRRRRRRRLRRRRRRRLRRRRRRRLRRRRRRRLRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR1R1R1R1R1R1R1R1R1R1R1R1R1R1R1R1RSRR1SRRRSRR1SRRRSRR1SRRRSRR1SRRR1R7R1R1R1R7R1R1R1R7R1R1R1R7R1R1RR1S1S1S1R1S1S1S1R1S1S1S1R1S1S1S17R7R7R1R7R7R7R1R7R7R7R1R7R7R7R1RS1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S11S717S711S717S711S717S711S717S71S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S171727172717271727172717271727172S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S17272727272727272727272727272727227S127S127S127S127S127S127S127S172727272727272727272727272727272S221S227S221S227S221S227S221S2277272727272727272727272727272727227S2272127S2272127S2272127S22721727272727272727272727272727272722227S2272227S2272227S2272227S227RRRRRRLKRQLKK$KLQLRLRKRRRLRRRLRRRRLRRRRRLK)D#C#C#KKKRLRLRRRLRRRKLLRLRLRKK#D"#CC">"DD#KKKRLRRRLRRSRRSRK*CD"DJKJD#C"#"C#C#KKRRLRLRRRRRKKD##KnnKJKsKKDC#C"CD*RSRRRRLRRLK#CJnK"D"D"ntstnKD##"DKRRLSLRRRKJ#nKtDJnKnJJos#sosnnD#JLRRRRRLRK#KnKsDKsKnKDsECtststsD#KRRRRRRKKDstKsDsnQosDJCJosntnttD*LRRRLRKDQoKQnKnKsnQnDJJtsttstnKJKRLRRR*KnssosKKsLsoJJtDostntnsnDKRRRRLJntKtnQnKsKstCDDJssoQKKsQKKRRLRRDQosnQntJosKnKDRsotnQKnKnKKRRRRKKsntntsoJsKtts#tnsstnKKstnJLRRSKnstsososKnKKntDQLsoQntttntKRRRRKnKKsstssKnQtKsDQntsnttsnKsKSRRRKKKKKttntKKnKsKKKoQonssntsoKtSSRLKnKnsstsnKQLRnJKQnsstntnQnQRRRRRRKKsKnKKoJLKJKKKKtntntQosnKSRRRRRRRLLQnssKKJnKKEQntstnsnsKRRRRLKLLRRRRLRLKKKKKKJtnQntsostKRSRRRSRRLQKKRRRLKKKKKKQKnKnstnsLRuRLRRRRRRD#KRRRRSRRKKnKJKQKsnKRRRSRRKKRSLJC$RStRRSRtLKKKnKnKKuRRuRLRRLRRRKKDQRSRKtSRSsKKKKJKRRRSRSRSRzRRLsKD*KRRYRRRRRnKsKKKRRRSRRRLRLLRRKnJDKLStSKRRLRKKnKKRRtRSRRRRLRRRLsKD#QRSRSuRRRKKKKsSRLStRRKRRRRRRnKJ"ERRLRRRRRDJsnKRSRRRSK*LuRSRLJKnJ"ERRRRRK#DKnKRSRtSRRRuRRRRKREmKJD"DK*KEC"nsKRtRRRLRLRRRLRtRRKKKJD""="C"DKnKLRRSLRRRRQLRRRRRLKJnKnJDC#DJnsKnRRRRtRLRRRLQLRLRRRtKnsKnQKsKKnKRRLRRRRSRQLRRRRRRRRSRKKnQnsnnnKRRRtSLRRLRLRRLRRRLRRRRRLKKKKKRRRLRRRRRSRRRLRKLRLRRLRLLRRRRSLRRRSRMRRSRLRRRRRRRRLRKRRRRRLRRRRSRRRRRRRRRRRLR2227S2272227S2272227S2272227S2277272727272727272727272727272727227S127S127S127S127S127S127S127S172727272727272727272727272727272S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S17S727S717S727S717S727S717S727S71S1S1S1S1S1S1S1S1S1S1S1S1S1S1S1S17R1R7R1R7R1R7R1R7R1R7R1R7R1R7R1RSRR1SRS1SRR1SRS1SRR1SRS1SRR1SRS1RR1RRR1RRR1RRR1RRR1RRR1RRR1RRR1RRRRRRRQRRRRRRRQRRRRRRRQRRRRRRRQRoRLRoRLRoRLRoRLRoRLRoRLRoRLRoRLRQRQLQRQLQRQLQRQLQRQLQRQLQRQLQRQLLQoQLQoQLQoQLQoQLQoQLQoQLQoQLQoQQKQKQKQKQKQKQKQKQKQKQKQKQKQKQKQKnKnKnKnKnKnKnKnKnKnKnKnKnKnKnKnKKKKnQKKKKKKnQKKKKKKnQKKKKKKnQKKKmKmKmKmKmKmKmKmKmKmKmKmKmKmKmKmKEmEmEmEmEmEmEmEmEmEmEmEmEmEmEmEmmDmDmDmDmDmDmDmDmDmDmDmDmDmDmDmDCgDgCgDgCgDgCgDgCgDgCgDgCgDgCgDggCmDgCmggCmDgCmggCmDgCmggCmDgCmgCgCgCgCgCgCgCgCgCgCgCgCgCgCgCgCgafgCafaCafgCafaCafgCafaCafgCafaCCfC`CfC`CfC`CfC`CfC`CfC`CfC`CfC``C```C```C```C```C```C```C```C``=_C`=_C`=_C`=_C`=_C`=_C`=_C`=_C`___`_`_`___`_`_`___`_`_`___`_`_` "--&TNPP &\Solution.htm#oneVBD:\Mth03\Trinom\Solution.htm#fourVBD:\Mth03\Trinom\Solution.htm#fiveT@D:\Mth03\Trinom\Solution.htm#sixP<D:\Mth03\Trinom\Solution.htm#2XDD:\Mth03\Trinom\Solution.htm#threet `http://www.tc.cc.va.us/vml/Mth03/Trinom/tri7.htm/ 0DTimes New Roman8bbd 0b & 0DComic Sans MSn8bbd 0b & 0OB DSymbolans MSn8bbd 0b & 0O0DArialans MSn8bbd 0b & 0O"@DArial Unicode MSbbd 0b & 0O"PDArial Rounded MT Boldd 0b & 0O" ` .  @n?" dd@  @@`` LDdg       2$?3iIuI3$$$b$ ʍ hGB%:3b$\Mnf'Konb$ 5_VW b$d(@v>K?hb$8LvƨͿ=b$|t?Ct-H^tc 1? 33B@B8uʚ;2Nʚ;g4MdMdd 0bppp@ <4!d!dg? 0b$b<4ddddg? 0b$b<4BdBdg? 0b$b f^___PPT9@ ? %)5Factoring Trinomials of the form ax2+bx+c with a =1.<6$C$K$C$ Dr. Marcia L. Tharp 8How to factor a trinomial of the form ax2+bx+c when a=1B9)fnf*To factor a trinomial means to write it as a product of two binomials. For example x2+6x+8= (x+2) (x+4) two binomials Remember we can use F.O.I. L. to multiply out (x+2) (x+4) we get: ctoring trinomials is an important Algebraic skill. It is used in higher-level mathematics and science. Besides that you will need it to simplify some polynomial expressions and solve more complex equations. fZZjZFgc gog gBgcGgcgc8"*'1"'om\Solution.htm#fourVBD:\Mth03\Trinom\Solution.htm#fiveT@D:\Mth03\Trinom\Solution.htm#sixP<D:\Mth03\Trinom\Solution.htm#2XDD:\Mth03\Trinom\Solution.htm#threet `http://www.tc.cc.va.us/vml/Mth03/Trinom/tri7.htm/ 0DTimes New Roman8bbd 0b & 0DComic Sans MSn8bbd 0b & 0OB DSymbolans MSn8bbd 0b & 0O0DArialans MSn8bbd 0b & 0O"@DArial Unicode MSbbd 0b & 0O"PDArial Rounded MT Boldd 0b & 0O" ` .  @n?" dd@  @@`` LDdg       2$?3iIuI3$$$b$ ʍ hGB%:3b$\Mnf'Konb$ 5_VW b$d(@v>K?hb$8LvƨͿ=b$|t?Ct-H^tc 1? 33B@B8uʚ;2Nʚ;g4MdMdd 0bppp@ <4!d!dg? 0b$b<4ddddg? 0b$b<4BdBdg? 0b$b f^___PPT9@ ? %)5Factoring Trinomials of the form ax2+bx+c with a =1.<6$C$K$C$ Dr. Marcia L. Tharp 8How to factor a trinomial of the form ax2+bx+c when a=1B9)fnf*To factor a trinomial means to write it as a product of two binomials. For example x2+6x+8= (x+2) (x+4) two binomials Remember we can use F.O.I. L. to multiply out (x+2) (x+4) we get: ctoring trinomials is an important Algebraic skill. It is used in higher-level mathematics and science. Besides that you will need it to simplify some polynomial expressions and solve more complex equations. fZZjZFgc gog gBgcGgcgc8"*'1"',2  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgijklmnopqrstuvwxyz{|}~sTo get this trinomial into its factored form we use F.O.I. L. in Reverse on x2+ 6x+8.ptR'''/'  Remember that the FIRST term x2 came from its factors x (x). So these two factors become the first terms in each binomial. r#''''/''('!''#'#M'#'""@8To do this we look at the miPowerPoint Document(h$DocumentSummaryInformation8L Root EntrydO).V) PicturesCurrent User)SummaryInformation(TR&"  !"#$%&'()*_TCCTCC ՜.+,D՜.+,T    $ On-screen ShowTidewater Community College$P 2 Times New RomanComic Sans MSSymbolArialArial Unicode MSddle terms of x2+2x+4x+8.9.'/''''''q When we use F.O. I. L. to multiply the OUTSIDE and INSIDE terms in (X+2) (x+4) r)'''''% % %+Lets make a table to organize what we know.*,+f" ?Interactive FOIL Exploration Print this page.,@ BAReview To Factor A Trinomial ax2+bx+c when a=1 Print this page.ZBBPractice Problems KNow click on the door to go on to How to factor the form ax2+bx+c with a>1TL#BbBJBm 0#:  ` f [f f3̙` 999f~` ___` 33f3f` f33f 3f3̙` 33f̙3f>?" dd@(33?Zd@ 33 d " @ `"  n?" dd@   @@``PN   @ ` ` p@@ TL <(  <|T  < " <  fgֳgֳd?P < 2  <  `gֳgֳ? < 2 Z < s *A?8 < # l1 ?"   T Click to edit Master title style! ! <  fܻ1 ?"`  ,  < # l1 ?"  >*  < # l1 ?"   @*  < # l1 ?"  @*N < 6޽h? ? f [f f3̙ Lock And Key  w@ @ (  @|T  @ " @  fgֳgֳd? < 2  @  `Wgֳgֳ?`  < 2 Z @ s *A?|  @ # l,1 ?" U   T Click to edit Master title style! ! @  ftG1 ?" t  W#Click to edit Master subtitle style$ $ @ # l<1 ?"0   >*  @ # l`1 ?"    @*  @ # lL 1 ?"  @*N @ 6޽h? ? f [f f3̙  0 6(   x  c $4@ U   ~  s *D@ t  H      !#$%N'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMOPQRSTUVWXYZ[\]^_`abcdefghij  0޽h ? f [f f3̙\     (    s * W< <$D 0     s *J<`<$ 0    C XA@D:\Mth03\TRINOM\TRI3\IMG001.PNG  ,$D  0%   `0N331?p @,$D   0 aX2+6x+8> 233 3333B @  `D33>? ` ,$D  0B @  `D33>? P ,$D  0B    `DB>? ,$D  0B  @  `DB>? ,$D  0B    `D33>?@ P ,$D  0B    `D33>?