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# Class Note for MATH 1300 with Professor Flagg at UH

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Date Created: 02/06/15

3 Math 1300 Section 41 Notes Greatest Common Factor and Factoring bv 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 a12 1y19 I x IL lVC x X 3XL 3 VL I b32 X31 i 7 syn 5L x l 4 X3 4 x 87 019 1 gtltt i d45 I X6145 lXLfs 3 x 45 3 V 5 5 V6 51Y 7 In the above the number 19 is an example of a if C 1M7 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 a and 3 s4 56 wwi G 7 54quot 3 39 Aquot 5 Ea 3 HQ3 3 2 5 b15and60 Co 5 A 9 0 3 5 23 as Math 1300 Section 41 Notes c2142and63 2f 39 as Lt G3 4 9 2 397 7 oazlBI 3 7 3 d 28and39 37 9 3 3 292 4A1 3633 373393 91 cape I 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 Examples 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 monornials as products of is without using exponents x4 7 I 7 7 X i 39 X 39 X 39 X Since eac monornial has 4 cs in it the GCF of the variables is x4 Step III The GCF ofx4 andx7 is 13 2 Find the GCF ofxyZ andxsy 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 f xsyzt xxxxg yy 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 f i 52 2 9 2 I Mathl300 LK Q g 3 Cf 3391ction4iNotes 1 a 1 3 G C F t 3 3 Find the GCF of24x4y and 9x7y4 Step I The GCF ofthe coef cients 24 and 9 is 3 Step 11 Re e the two monomials as products of 1s and 2s Without using exponents y GCF XXXX xy xxx yyy Since ch monomial has 4 cs and l y in it the GCF of the vaIiables is x4y Step III The GCF of24x4y and 9x7y4 is 3x4y 3 Find the GCF Ofl41 and 21x32 V l a a 45 l 4 A E 2 W YL 1 5 1 3397 C 7 GCFgtlt y Xt cc 5 7X7 4 Find the GCF of18c9 and 6c x V lo r 5 I G 33 Ce CCC CC 2 5 C t x I gt 23 gijwjx3 GCF ccC CClt o GCF GCF z 904 5 Find the GCF of23a6b3 and 42a302 23 41 53 I aaQZo L rmb turf CCle a3Cz GCF0 GCF 111 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 undistIibuting the GCF out of all the monomials in the polynomial Note that if the leading coef cient is negative mp9 then the GCF should also be negative 71 W P am Lt gt 45a 5 Mia ygnmmty Math 1300 Section 41 Notes Examples Tarms 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 2 Factor 8x3 6x2 3 Step I Find the GCF ofthe coef cients GCF8 6 2 a 39 9 w 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 3X 1 8x36x22xz Z 3 z z2x4ia 2 QXIXTX LX31X 2x 2x Q2 9X 99 3 Factor l8y3x712lyx7l 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 igyxo 1 2lyx7 1 2 9 m 6 Factor 3ab7b I 3 17 QCF l E Variables ab 5 QCFZE ToluL GCF t b 3mbl 75 b3 Jr b3 7gt 92 3 X 7 Factor 48113117 7 36a2b5 Wm 395 7n use 390 M farf o GCD I NM 35 36 29 7 c F92 36I133 Luau QCF 64 L Ivmdjob ask 03435 135 39 390 Wu H Sac 39oxlb g TM GCF 4alb 3 quotI 1 539 1 65 A29a3 7 3 ampb 9580 b f 3 a a 46L v 39 qmb JL azb quot40367750U 6b N M i 8126c glare A GCF Lf fag5c 20 c Variadafes ogiac abcm ELI Gang45 4abclt aggzg g 4 ltxr7xL a7x7 NutquotJIM GCFl l27 VA GF 3 x X5 x or cs x7 4 5 z X x gtlt w arw 13xV9772 ti3x Ian3 In COMum 9 3 X 3 3 WM oh Q39yz M51 57 haven1 con rm NW 27342 4033 6 3 l 5 Z VM Aquot VA 73 GC F V V 9 C0 mmon 5C F E 3 L 3 Tm L magma GHXX 1 34 Ci 3X 9 72773X 397 67 3X3 g 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 Factor xzy 6x 3xyZ 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 ny 6X KM 3W2 18y 3mm 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 2 Factor 2x3 3x247 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 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 E 122337 KR 7 Lf 1a3x 394 3x 3 L39 4 53 3X gtaa L 4 x V l36ax6xb 0ay10yb gay 6Xbgt G Wf l oolt 6Xb 6X GCF 4X 6gtlt Gamma 0 HO A 10 A 340 Oy7 ms oypk b cxauo 4on 01 40 6x 207 14 6x 9xy10y15y2 gEoUP c g 4 6x fx 3 32lt 3 7 git 3x41 37 GCF3X O 5 5 E W 3 GCF 53 5

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