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Fundamentals of Math

by: Alvena McDermott

Fundamentals of Math MATH 1300

Marketplace > University of Houston > Mathmatics > MATH 1300 > Fundamentals of Math
Alvena McDermott
GPA 3.69

Mary Flagg

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Mary Flagg
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This 30 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1300 at University of Houston taught by Mary Flagg in Fall. Since its upload, it has received 8 views. For similar materials see /class/208402/math-1300-university-of-houston in Mathmatics at University of Houston.


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Date Created: 09/19/15
Math 1300 Section 24 Notes Equations ofa Line Forms of Lines 1 2 3 The standard form of a linear equation is given by Ax By C Where A and B cannot both be equal to zero The pointslope form of a linear equation is given by y 7 y1 mx 7 x1 Where In is the slope and the line passes through the point x1 yl The slopeintercept form of a linear equation is given by y mx b Where m is the slope and b is the y intercept Find an equation for the line with a slope of 12 and a y intercept 2 m V in T 2 Slept in 39FOV M v r i X i 2 Li t x L lf lrd39e 9 k 5 3K 4 5 poiq l 39 SO LL56 P0124 glotpr F O Wl Wlopt 34 PM 3143 gt9 3 VHV1 4Q XQ V Q15X3lgtl y ce 34 lt33 Vp Math 1300 Section 24 Notes 4 Find an equation for the line with a slope of 6 and passing through the point 0 5 quot gtlt t 0 cf t S m t PM Rm Lf y r M Y 39 y y 5 r 4 x o w S Q X 5 Find an equation in slopeintercept form for the line that passes through the points 5 2 and 3 2 2 b q I We it r 7 2 Fofai S 096 quot Z39i Fi V Sicpc g O quotEigtlt 33 A749 X393 Whale 2 7X 1 gum SUBTICAC39T g t 1 2LX quot 1 6 Find an equation in pointslope form for the line that passes through the points 5 2 and 3 2 Sowv1 610 M F04 rd 5 1906 I y 4 3 v 92 gtlt 3 AXQVY QC 5W 34m wH k Ex 539 M the in s opt 14Vde gm Kyquot x 3 igtlti Tix Slope m kl L77 Q Xix gs27 Q Y 1 L1 0 7VO KWA E 39 rS THE 34 a x vim JW 7 39 s0 e Cr7 o F V 3 t 7 70 Poinlc Sope Form POIN K 31 1 59oe 7 0 0 1 Ogtlt 33 g I a O ybquot Math 1300 Section 24 Notes 10 Find an equation in slopeintercept form for the 1iney 7 3 5x 6 in point slope form 3 5 x A F C Szope 13 a C7 0714 V L04 3 4 3X 139 3 0 3 3 S1 ope 5 n 139 S 1 1 Find the x and yintercepts for the line 8x 7 2y 16 Write each as a coordinate point 12 Find the x and yintercepts for the line passing through the points 3 3 and 2 1 Math 1300 Solving Equations by Factoring Definition The zeroproduct pro a0orb0orboth Definition A quadratic equation where a b and c are numbers an Solving Quadratic Equations To solve a quadratic equation we 0 0 Factoring is usually a help move all nonzero terms to one sir use the zeroproduct property Examples 1 Solve the equation x2 7 5x 7 Only the lefthand side LHE of terms is necessary Facto Factor ZPP x Solve for x 2 Solve 2x218x772 0for All nonzero terms are on the Factor 2 39 x Solve for x J 3 Solve the equation 6x2 7 27x This equation has nonzero tlt possible to solve move the 7 6 Factor 31 Solve for x 4 Solve 16x2 71 Subtract 1 from both sides Factor 4x 7 Solve for x 5 smug 0 1X 16x2710 10 or 4x10 x 4 or xquot4 x3 O x7gt0 1 Mix 73 a Raw 41 2 12 4 32 3X 3 M web 0 4x 7 14x 1 0 Note This is a special polynomial Math 1300 Section 44 Notes 5 Solve 712x2 7 717x 6 This quadratic equation has nonzero terms on both sides of the equal sign so we must move all the terms to one side Add 17x 7 6 to both sides and solve 7122 17x 7 6 7 0 Factor 71 4x 7 33x 7 2 0 Note 71 0 ZPP 4x730 or 3x720 Solve forx x 34 or x 2 6 Solve for x xx 7 2 71 Make one side equal zero xx 7 2 l 0 Distribute x2 7 2x 1 7 0 Factor x 7 l2 0 Note This is a special polynomial ZPP x71 0 Note Noneedtowntex710tvvice Solve for x x 1 Solving Other Polynomial Equations Solving other polynomial equations is done just like the quadratic equations Set one side of the equation to zero factor use the zeroproduct property and solve for x Examples Padre r C C F 0 14 X 1 Solve4x316x215x0 XLfYL E 6X l9 0 This polynomial is already equal to zero so all we need do is factor use the ZPP and solve for X 4x316x215x 0 Factor x2x52x30 ZPP x0 or 2x50 or 2x30 Solve forx x0 or x3952 or x32 2 Solve x3 2x2 99x Get LHS equal to zero x3 7 2x2 7 936 0 Factor xx 7 llx 9 0 x0orx7ll0orx90 Solveforx x0 orxll or x79 3 Solve 7x3 714x2 7 0 Set one side equal to zero Already done Factor 7x2x 7 2 0 ZPP 7x20orx720 Solve forx x 0 or x 2 Math 1300 4 Solve x3 18 2x2 9x by factoring 23 Set one side equal to zero Rewrite Factor ZPP Solve for x 5 Solve 30x3 7 3x2 7 9x Set one side equal to zero Factor ZPP Solve for x Section 44 Notes 1872x279x0 72xl79x180 x279x72o x3x73x720 x30 or x730 or x720 x73 or x3 or x2 0 Alreadydone 3x5x732xl0 3x0 or 5x730 or 2xl0 x0 or x35 or xquot2 6 Solve 2x 712 2xx2 7 20 8x2 2 Set one side equal to zero Simplify Rewrite Factor Z PP Solve for x Solve the following equations by factoring l r 0 2x71272xx272078x22o 2x272x172xx272078x22o 2x2 4x272x340x78x2720 72x376x236x0 72xx6x730 72x0 or x60 or x730 x0 or x or x3 54 5d owl nms an 039 54 2 cm out cF 3 i O 967 m Sap s Faxt 682243143 5h l chjrra we a 55 4 Eero39ProM PM at cl ac 5f 5 5olve b 5 74 racLOKJO a X I UCog wo NLMLRM M Q b quotgt Pr eiim 3X 23 IS gt 5 S Fi Evero PFQM 3x QBXDO MW 3xlO n gtltHl 30 3 Stth 3X L Math 1300 2 xjix 4 313 8 3x PM VM 5 823 gxllux 399 U 39 39 8 C3 ac r7 1quot0 2Llgtlt L 9 10 Pm a g F L 3 141 6 12 3X gawk PW ax 3347lt o Mathl300 Zero x 43 0 qx I OrSection44Notes r 6 3x22x8 L f Jr lx 3 3XL QX8 O PW 0L3 ct g ac1l f bt1 P924 L6 X g Pro PM 3x LO 33x 9 PM 33ltthlt930 7 4298 148x2100 BXEBT gg a M 7 2 x7 3 a H rlgtltagtlt 74c 4 Ah 9gtlt 3x 4 337x gt zero 2 3X C 4 E7y1 IYK 56X XO 7 a C 3 2 3X 41 9 X ti Pg X o P T 7x Q 8 ssx 32x2 2 39 i 7 7 31 31 X 1 X 7 8XL 321th O GCF t lt5X7 ng XL L0 gxl gtlt Q 0 O Math 1300 Section 44 Notes 939 45 o 4x Mlwa 40 o 46x1 IIX b 0 06 ELWCi 1 Tllquot M o 0 l X LMX Dgtlt i019 0 b gtU0 Wain 75 9 10 410 CO GCF lt39 xtm 7x 39397 wwa 7 W 7 X 7 X 3 X2 l 11 06x27x720 7 X 33 x Dlt I100 GCF LNW M 10 ZUO x3j xl0 X l0 ox 10 Q0 PEI 7 3 if H QLO Xeo K 3 x X I 4a XI VIEW 43925 O algxax 5 3x cg4 th50 77x72 728 3 2 6 7gt l4gtltLU quot23 1quot 5 7x39L 4h2W 17x1 ex ch 7x 4 z X 7gtlt x h 37x 9 H W 939 M 6 Math 1300 hapter The Real Number Field Part 2 Rational and Irrational Numbers Using only integers we do not have a number to describe dividing one whole pie into four equal pieces The rational numbers were created to solve this problem A RATIONAL NUMBER is a number that can be represented as a ratio of two integers a E for any integer a and any nonzero integer b Note that a over b and a divided by b mean the same thing solving the division problem is built into the de nition of a rational number Note also that every integer is also a rational number since aninteger z can also be Z expressed as I z dividedby 1 Now that we know what a rational number is how do we add subtract multiply and divide them ADDITION and SUBTRACTION Adding and Subtracting Fractions 0 Find a least common denominator using method for LCM 0 Change the numerators of each fraction 0 Add or subtract the numerators keep denominator unchanged 0 Reduce Examples 7 a 9 34 2 539 1 F27 5 54 SH 54 Z 7x3 g 1 2x2 9 r 9x3 4 9 7 47x2 5 7 5 Am 804L412 rgaxa2x3x3 73 Z 973X3X3X3 4cm 0239 gtlt 3 532M z 54 0 A91es7 90uJe 111 97 S 05L Alma 1 33 73 Anemia LCM g ix3 M gtlt 3l3S 1 1 3gtlt3gtlt33Q LC 3 lll3933ZZ 4955 Tg 53 3 13539 2 7 27x5 35 MULTIPLICATION and DIVISION Multiplying Fractions 0 Simplify the fractions if not in lowest terms 0 Multiply the numerators of the fractions to get the new numerator 0 Multiply the denominators of the fractions to get the new denominator Examples 1 X 2 1 1 x z 53 5x3 qg urn mcquotm a VQ Dividing Fractions Multiply the rst fraction by the reciprocal of the second fixagnipl es 3 7 r 3 X 7 XY 7 1 4 waives 2383XLM 9 1 4 C7 9 7x 39 iL J i 9X 2A ex 2 I 3 POWERS 7 SHORTHAND for multiplication and division 0 De ne r a0 1 W Er ower 5 1 1 WW 0 a 45 o 2 2 7 7 anaaaa 023 ix C 1711 3x3gtlt3 X3X3gtlt313 a ni 1 CAL a D E 3 l The powc pudes wong 100 of W M Va qbks fCe gt Cxgc 35 23x232 20C 33 I Z 3 5 4 2 31 3 CQampJk L3 327 Camp Mmtdjw m 3z jUS are M 39 Medas 3 Ewen 1 spsm g2 t ISSUES Ma GHQ 52 em z E 39b 3C c L 395 amp 0 4 7 2 a 3A 6 6 t t zgezj 739 26 CL FL L g e 1 N S 2 i j 52 I 1 5 39 5 2 A MfrIt I I Piy on 0 O39HLV Srdc 39 op Cnacwtz39w OHMQ 51 was Me 049 e m an 90 or has140m Itavxe 0 Y A 3965 place quot lt gtlt X 1 O ELT7 33 WW 1 M H over 9 213 3 if 3 g r 397 X Q Ma 2 3 1 l 37V 72 3V 37 t 2 7 V11 2 X 39125 quot 3y1251 3 Byz g Different Ways of Expressing a Rational Number Proper and Improper Fractions PROPER FRACTION IMPROPER FRACTION I To WM 953 TOP W 5 579 E n 5 7 1 1 5 527 MixedNurnbers2 6 10 3 i 3 9 2 4 A a L L 1 2 3 I 2 I 1 Q3 u 3 3143 i O fQiZjJ O Q 1 39 a 42 t E112 gj 7 7 f I U I a RM 0 MAL Null18565 NCLUDE m MED mumerzs Decimals 7 Fractions meet the base ten number system 2 v 1 Ob 2 039 0107100 gfmlqakmj 7 0 034532 34532 D 3 3 Wb 100000 E arrow31 0 A u m BEES How do you turn a fraction into a decimal Divide With long division 2 O 9 Q 25 3WD 25 t a 98 539 0 1 Q Q g 0 3 3 3 033332 amp0 3 o o To iiiii g i ilmmsu e 1 Notation J3 Or r 9 3 Q up 444449 2 5 2245555 quot Q5 5 An example of a decimal number that goes on fore7erbu5tdoes not repeat LQWW2444444I I R7 I Wo quot CNNlab Od thinalN b 071241 Tlrieeexariiessiinox lt y ilsIreqilrijlalent to saying y 7 x gt 0 72 How to compare two rational numbers 1 In Decimal Form 04443252304433324 3445 3442 1053 lt 100 0332 lt 10004 5GGEL 4 gt 2 C 011394 2 In Fraction Form a Express both numbers as fractions 7 proper or improper b Find the common denominator c Express both fractions as equivalent fractions over the common denominator a b d Compare the nurnerators lt 1f and only1f altb C C y v Practice L lt 23 lt 42 lt5 1 9 QI U 7 13 ET 8 3 39 LCME H xj 41 LCMaLK Z 5 1 X F t z 1quot LCI WQS3 x i 3 7 a 3 g 833 i ri Xquot 32 3 1 33 gcf 3 33 2739 4l1lt 45 2 2 lt 30 2 5 I 6 lt 397lt 5 3 a 5 l A 5 9 0 f O 3 0 Summary We now have the rational n ers and we can perform all four operations with them and get a rational number answer So we have all the numbers we need OR DO WE Problem 1 Given a square with sides of length one inch how long is the diagonal Q 7 r 1 7 I I v x igt x V a E Problem 2 Find the ratio of the circumference of a circle to its diameter c wamp Neither of there two problems have answers that are rational numbers We need more numbers to describe REAL PHYSICAL LENGTHS Irrational Numbers are real numbers real points on the number line that CANNOT be expressed as a ratio of two integers fraction Examples pi the square root of2 We Will work With square roots in this class Other classes Will introduce you to other irrational number N Simplifying Radicals M F M hominich mam w H r t u dg w Nl R A PampCm4 Note When there isn t a number for n it means there s really a 2 there I a I W Examples 3 w I 39 ca rm 2 quot W 3 1 g A a WOO g2 00 32 4x426 XIO ilra 344 4 1300 3 L 7L W 10 A 39 qo 0 3J5 4 M t 2 K64 977 agitaxag 2 grT JQQXKQE39JE 3992 a xjlx lx 3 399 4 r 9x02er axax X31 7 2 41 2w r 5142 W l V32 4xf fl XQX Pigec kzsvls Y CL 51 t 13 am n7 Ram 61 Q 2 Kirk PosiHV nszmlxek 2 m s fr QJHOV O L Section 17 Notes 8 1 e gt My Solving Linear Inequalities An inequality is similar to an equation except instead of an equal sign you find one of the following signs lt S gt or 2 Now gt and lt are strict inequalities and 2 and S are I inequalities that include equals read these last two signs as greater than or equal to and less than or equal to respectively The solutions will be different because a linear X C X equation has one none or many solutions A linear inequality has a solution that is over 39 an interval and the answers are in what is called interval notation This ensures that you A Whave the complete answer H if i In this section we will only be concerned with real number solutions reor 9V x Interval Notation Inequality Interval Notation Math 1300 x lt A 7 00 A The rounded bracket means that A is not included x g A 7 00 A The square bracket means that A is included x gt B B 00 g E 9 x 2 B 3 00 lt X A lt x lt B AB A g A g x g B A B A X 6 W 1 1 dth N b L39 5 A 5 t nervasan e umer 1ne wig XltH keep Wn e gorQVe V 00 H 26C A I PooML parqdf x w A Ismail1C Go j w b m cw x3 gtltgtB 68 X B 3500 Procedure Get X on onje side by imelf or in the case of compound inequalitites get X by imelt in the middle Operations you can use to get X by itself 1 Add or subtract the SAME real number on either side all sides 2 Multiply both sides by the same POSITIVE real number 3 Divide both sides by the same POSITIVE real number Multiply or divide by a igative number SIGN Math 1300 30 X lt 3 Section 17 Notes Note If you multiply or divide an inequality by a negative number you have to reverse the inequality sign for the solution to be correct For examp e 7 4x gt 12 7 ix gt2 0 4 4 313914 75 x g 73 lfwe don t reverse the sign the answer will be incorrect Check the answer by testing x 1 and you get 7 471 gt 12 4 gt 12 So 1 gives the WRONG answer If we reverse the inequality we have x lt 7 3 Check this answer by testing 5 7 4 is gt 12 D 20 gt 12 3 K So reversing the ugquality gives us a TRUE statement 3 Qlt Simple Example 2lt5 but 2gt 5 9013 54 75 47393391 10 Q3 45 Examples 1 Express the solution of 7 4x lt 8 in interval notation fx 3 1 if X 72 CW 72 so Math 1300 Section 17 Notes 2 Express the solution of 5x 7 1 g 7x 5 in interval notation 6 lt 5x4 S 7x 5 1 Q f 5x 6x lt O i 3 2x74 3 X 73 O Q S QX 3 I M 6amp9 Key X in WW Goa A ltgtltlt6 QltgtltLflt5lz I va Li 39q C 01 5 S Olt 125 S g 3 0 DC 5x 3x 6 lt QX39F Dll i39ribt ka Likeij ax 9 4 ax 7 QX 2gtlt C lt C I 4x f s X39S weka aidtfywemsuif7 slcd W I 5 ruz j X 5 EVERY 65m JENe rchL C as 6 Express the solution of 7 3 g 2x 1 lt 7 in interval notation 7 3 Qx 4 lt H l l 4 s 51 2 Math 1300 6 C l 0 23 b gt i in interval notation l 2 6 324 S 7 Express the solution of g 2 H 13 L I I Q Tarn L We 8 Express the solution of 4 7 3x g 71 8x in interval notation Ll 3x lt L Flu umauei sic 00 are o Jim 5 ZSOLUTKIA on 17 Notes Math 1300 Section 41 Notes Greatest Common Factor and Factoring by Grouping Review Factoring De nition A factor is a number variable monornial or polynomial which is multiplies by another number variable monornial or polynomial to obtain a product 1 List all the possible factors of the following numbers a 12 I YISL gtlt 7 136 39 Y 3XL 3 VL I b32 X31 i 7 gym 6L x m 4 X3 4 x 87 9 Md 1 gtlttr d45 I qug lXLts 3 X 45 3 X 5 Eve u V In the above the number 19 is an example of a if i Mi number because its only positive factors are one and itself Review Greatest Common Factor De nition The greatest common factor of two or more numbers is the largest number that divides goes into the given numbers with a remainder of zero 2 Find the GCF greatest common factor of the following numbers 4 3a6and5 3C 5 4 339 39 3 GCF 9 3393 3 2 5 b15and60 Co 3 A CD 0 5 2 3 a Math 1300 Section 41 Notes 0 2142and63 2 l as 1 1 63 a 7 4 2 399 39 3 7 a 2 63393quot 3 7 3 d 28and39 CC 3 7izt a 3 3Qx2x7 4 313 3 3393 5 1 cap I nua W Greatest Common Factor of Polynomials In order to nd the GCF of two or more monornials I Find the GCF ofthe coef cients 11 Find the GCF ofthe variables III Rewrite the GCF as a product ofthe GCF ofthe coef cients times the GCF ofthe variables Exam les 1 Find the GCF ofx4 andx7 Step I The only coef cients are 1 s so this is the GCF ofthe coef cients Step II Rewrite the o monornials as products of is without using exponents x4 7 x Since eac monornial has 4 cs in it the GCF of the variables is x4 Step III The GCF ofx4 andx7 is 13 N Find the GCF of xyZ andx5y4 Step I The only coef cients are 1 s so this is the GCF ofthe coef cients Step II Rewrite the o monornials as products of 1s and 2s Without using exponents Z Xy 5 4 Since each monornial s c and 2 2s in it the GCF of the variables is xyZ Step 111 The GCF ofxyZ and x5y4 is xyZ 7 aka2 3 Ection 41 Notes Lf Math 1300 Q g 3 Cf L QCF3 1 a 1 3 Find the GCF of24x4y and 9x7y4 Step I The GCF ofthe coef cients 24 and 9 is 3 Step II 39 3 FindtheGCFofl4a1and21x5yZ V infoc l 2A7 14 7 w r M 33 XSVZ 39X X X39X g y F 3 7 GC 7 GCF X y x y GCF 7 X y 4 FindtlieGCFoflchand604x V a r 5 I e 3X6 Cel CC C CC 2 5 C4X i gtlt 23 lgj jx3 GCF ccC CC o GcFi GCF c 5 Find the GCF of23a6b3 and 42a302 23 41 53 I aaaZo L Emma luff GCF r ag GCF 0x C c GCF l39al Factoring To factor a polynomial an attempt should be made to nd the GCF of the monomials in the polynomial Then this GCF should be factored out of the polynomial by undisttibuting the GCF out of all the monomials in the polynomial Note that if the leading coef cient is negative then the GCF should also be negative Wquot M We 4a w Mia 5111 amt j l Math 1300 Section 41 Notes Examples Tafm 5 7X 7 1 Factor 7x 14y Step I Find the GCF ofthe coef cients GCF7 14 7 6 CF quot 7 Step 11 Find the GCF of the variable parts There is none so nothing changes Step III Divide all the monomials by the GCFs and rewrite with the GCFs out front 7x 14y 77 7x4r147y 7x2y 3 5 g 9 2 Factor 8x3 6x2 3 a Step I Find the GCF ofthe coef cients GCF8 6 2 a 39 X 7 Step 11 Find the GCF of the variable parts GCFx3 x2 x2 1 Step III Divide all the monomials by the GCFs and rewrite with the GCFs out front X 3 z z cF 8X 3 1 83x36x22x28xzi2 0W 6 2 xix Lia x x 2 2x24 1 2 lt1 I X 3 Factor l8y3x712lyx7l 9 Step I Find the GCF ofthe coef cients GCF18 21 3 Step 11 Find the GCF ofthe variable parts GCFy3x 7 1 yx 7 1 yx 7 1 Step III Divide all the monomials by the GCFs and rewrite with the GCFs out front 18y3x7l2lyx7li ljingriL m7 6iizii 6 Factor 3ab7b 1 NM 3 7 65F l valalgles 5L5 b Tom 6 CF b Balol Wb bK g jeg39 wof gt 7 Factor 48113117 7 36a2b5 WM 39sl 7n use 390 M o GCD I NM 18 3Q thzjn GCF226 36 v 33 Umlm chz 6Ll L I Vahdmc a3 97 051103 off a b Mn 11 GCLC alb5 new GCF 4a 3 30351 f 36albfz 4465 i T39ZS39QZ 67 3503 5 41 W quot4oquot 7wgl NW 3 a 3132 mm ramfa A GCF 4f 656 as c Vmaaares 1 bc4azog quot wk 9 a CcFmbc 4abc 1ampquot323 k Xr7Klt3627X7 Nume GCFl rll7 y l Vomch G Fx 3 X X9 X GCF xcrgti fl 5 x Z 97 K7CIXV3 amp7X X 6 F g L quotX X f 3 37y 3xnf iS x Ian3 I ComWA QBX WM oh Q39yquot M11 6 73 havefn co1141M NW 1 Vow AL 1 C0 mmon 5C F 7 3 7 My Tm 7 L 3v q393x 3 q3x3 a 5g Math 1300 Section 41 Notes Factoring by Grouping If a polynomial contains four or more terms it may be helpful to put the terms into groups of two and factor out a common factor from each of these groups This is called groum1g Examples 1 Fquot 5 Factor xzy 6x 3ch2 18y Step I Group the terms so that each group shares a common factor xzy 6x 3xyZ 18y xzy 6x 301Z 18y Step II Factor out the common terms from each group x y 6x x y2 18y 3y Step III Rewrite the polynomial as the sum of the factored groups xzy 6x 3xyZ 18y xxy 6 3yxy 6 Step IV Factor the resulting polynomial from Step III xxy 6 3yxy 6 xy 6x 3y Factor 2x3 3x2 2x 3 Step I Group the terms so that each group shares a common factor 2x33x22x 32x33x22x3 Step II Factor out the common terms from each group 2x3 312 7322 3 2x 3 12x 3 Note There s no common term so the GCD is 1 Step III Rewrite the polynomial as the sum of the factored groups from Step II 2 3x2 2x 3 x22x 3 l2x 3 Step IV Factor out the resulting polynomial from Step III x22x 3 l2x 3 7 x2 l2x 3 Factor 3x3 3x2 7 4x 7 4 Step I Group the terms so that each group shares a common factor 3x3 3x2 7 4x 7 4 7 3x3 3x2 4 7 4 Step II Factor out the common terms from each group 3x3 3x2 7 3x2x 1 4x 7 4 4x 1 Step III Rewrite the polynomial as the sum of the factored groups from Step II 3 3x2 7 4x 7 4 7 33 1 4x 1 7 33 1 7 4x 1 Step IV Factor out the resulting polynomial from Step III 3320 1 7 4x 1 7 3x2 7 4x 1 i mes Q24 K K quxy 394 3x 3 ask L0 an 10yb A 6ax6xb 6x GCF gtlt O flay Gar 1047 swam 6X 0 A O 10 W f 7037 scamo 4 10 M9 40 6x H07 Gioji xy 3 3x ilt 7 3X51 37gt GCF3gtlt o 5 1 gamma 0711999 557 gj F GCF5 35121 372 3X3375 57Q37 quot 37X 4 CL Gem ygt 37


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