MULTIPLYING OUT & FACTORING

igitl ircuit Engineering MULTIPLYING OUT & FTORING I IGITL SIGN Except for #$&@ fctoring st istributive X + X = X( + ) 2nd istributive (X + )(X + ) = X + (X + )(X + )(X + ) = X + Swp (X + )(X + ) = X + X The Most ommon Stupid Errors Using Not Using X. Y = XY YX + X = X X + = X Generl emorgn Y + XY = X + Y F(, b,... z,+,.,,0) F(, b,... z,.,+,0,) rleton University 2009 dig4fctoringh.fm p. 26 Revised; Jnury 29, 2009 Slide i Multiplying Out nd Fctoring Sum-of-Products, Product-of-Sums Multiplying Out Use Tke dul; Simplify & ; reverse dul; then use Use Krnugh Mp Fctoring Use 2 Tke the dul multiply out tke the dul bck Find F using emorgn, then use Krnugh mp rleton University dig4fctoringh.fm p. 27, Revised; Jnury 29, 2009 omment on Slide i

Stndrd Forms Sum-of-Products Product-of-Sums These re stndrd templtes or forms Every logicl expression cn be converted to either of these forms. Sum of Products Σ of Π, OR of Ns, S of P bc + cde + d + f + db Single vribles Ned together into terms. These terms re ORed together. Inversions re only over individul vribles. No brckets. Product of Sums. Π of Σ, N of ORs, P of S Single vribles ORed together into terms. These terms re Ned together. Inversions re only over individul vribles. rckets only round vribles in OR terms Questions Is b + cd + b(d+e) Σ of Π? Is ( +bc)(d+e) Π of Σ? (+b+c)(c+d+e)(+d)(f)(d+b) dig4fctoringh.fm p. 28 Revised; Jnury 29, 2009 Slide Stndrd Forms Terms Terms term in logicl expression is number of single letters Ned together. Exmples: bcd bcxq xy epending on context, it could lso represent number of single letters ORed together. Exmples: +b+c+y+q + Sum-of-Products These must be expressed lgebriclly s the OR of Ns. One ws not llowed to put in brckets, XORs, nor long inversion brs. Thus none of the exmples on the left below re true sum of products. NOT Σ of Π bc + b(+d) + de bc + bcd +bf (bc + bfg + bcd) c + bg d+bc bc These re Σ of Π fter finding n lgebric Σ of Π form, it cn be esily chnged to NN-NN logicfor implementtion. Product of Sums These must be expressed lgebriclly s the N of ORs. One must use single-letters ORed together into terms which re Ned together. There must be no XORs, nor long inversion brs. Thus none of the exmples on the left below re true product of sum. (+b+c)b(+d) (+b+c)(d+e)+d (e+b+c) (+b+c)(b+c+d)(+b) ( c+b)(g+h) +b+c These re Π of Σ NOT Π of Σ fter finding lgebric Π of Σ form, it cn be esily chnged to NOR-NOR logic for implementtion. rleton University dig4fctoringh.fm p. 29, Revised; Jnury 29, 2009 omment on Slide

Sum-of-Products Product-of-Sums Two Stndrd Forms Sum of Products (Σ of Π) bc + e + ce + bd +... OR of Ns c b e e c d b Σ of Π Σ of Π (NN-NN) n be implemented s NN-NN logic NN Product of Sum (Π of Σ) ul of Σ of Π (+b+c)(+e)(+c+e)(+b+d)(... N of ORs n be implemented s NOR-NOR logic c b e e c d b c b e e c d b Π of Σ NOR c b e e c d b Π of Σ (NOR-NOR) dig4fctoringh.fm p. 30 Revised; Jnury 29, 2009 Slide 2 Sum-of-Products Product-of-Sums Stndrd Forms oolen expression cn be represented in mny wys, see below. wy to see if two. There re mny templtes (forms) to define oolen Function. form tht cn define ll oolen functions is clled stndrd form. Some Stndrd Forms. b b The truth tble is very bsic form 0 0 0 Σ of Π expression. b + b 0 0 Π of Σ expression. ( + b)( + b) (fctored form) 0 Krnugh mps form tht will be introduced shortly. Stndrd Forms inry decision digrms, which will be used lter to build rbitrry circuits with 2-input muxes. Why we sy Σ of Π for Sum of Product n Σ is used for repetitive ddition, s in x i = x 0 + x + x 2 + x, n i = 0 n Π is used for repetitive product, s in x i = x 0 x x 2 x n i = 0. specil type of stndrd form is the nonicl form. It is one which is unique. If two pprently different oolen expressions re reduced to cnonicl form, they re the sme if-nd-only-if their cnonicl forms re identicl. Truth tbles nd unlooped Krnugh mps re cnonicl stndrd forms. Σ of Π nd Π of Σ re only stndrd forms nonicl originlly ment ccording to the rule or lw, prticulrly the church lw. rleton University dig4fctoringh.fm p. 3, Revised; Jnury 29, 2009 omment on Slide 2

Fctoring nd Multiplying Out Trnsforming Σ of Π Π of Σ Multiplying out Trnsforms Π of Σ Σ of Π ( + c)(b + + d) b + c + d Fctoring Trnsforms Σ of Χ Π of Σ b + c + d ( + c)(b + + d) Why Fctor? : Typiclly both forms hve bout the bout the sme size, but sometimes fctoring cn sve significnt logic. Σ of Π Π of Σ Σ of Π Π of Σ b + c + d + e = (b+c+d+e) 5 gtes, 8 letters 2 gtes, 5 letters cb + c b + cd + c d = ( + c)(b + d)( + c) 5 gtes, 2 letters 4 gtes, 6 letters c + e +bc + be = ( + b)(c + e) 5 gtes, 8 letters 3 gtes, 4 letters 2: Sometimes NOR-NOR logic my be desired (Fst fll time). dig4fctoringh.fm p. 32 Revised; Jnury 29, 2009 Slide 3 Fctoring nd Multiplying Out Σ of Π Π of Σ Σ of Π Π of Σ Why fctor? Usully the (Π of Σ) nd (Σ of Π) forms re bout the sme complexity, but chnging forms for prt of the circuit cn sometimes sve significnt logic. Logic minimiztion progrms will switch from one form to the other in different prts of lrge circuit. Π of Σ in NOR- NOR form, cn give fst results with flling signl =>0. 4-. PROLEM Identify the following s: Σ of Π, Π of Σ, or neither.. (W + Y + Z)(X + Y + Z)(W + X)(W + Z)(X + Y + Z)(W + X + Z) 2. (b + c)(d + ce)( + b + c + d) 3. dc + (bc)+ cb 4. bc + c(+b) + dec +b 5. + b + cd. NMOS trnsistors re fster thn PMOS. NOR gte use prllel NMOS trnsistors to lower its output signl. ll other things being equl, this leds to slightly fster circuits. rleton University dig4fctoringh.fm p. 33, Revised; Jnury 29, 2009 omment on Slide 3

Multiplying Out; Methods Multiplying Out hnge Π of Σ Σ of Π ( - - - )( - - - )( - - - ) ( ) + ( ) + ( ) st istributive Lw () x( + ) = x + x Methods of Multiplying Out I. Use the st distributive lw x( + b) = x + xb - This lwys works, but - it usully gives very very long result. II. Simplify (use 2, S2, Sw) OR tke dul (use, S, Sw) tke dul before using - Gives shorter nswers - Shorter steps to get nswer - Requires more thinking III. Use emorgn s Lw to get F Plot F on K-mp Then plot F - Esiest to do - Gives the simplest Σ of Π nswer. - Gets messy for 5 or more vribles. Exmple I ( + )( + + ) Use () = + + + + + 0 Use = 0 = + + + + Exmple II ( + )( + + ) = ( + )( + [ + ]) = [ + ] + = + + Exmple III F=( + )( + ) Use (em) F = + Use Mp F = + Use (Sw) (x+)(x+) = x + x Use () F F dig4fctoringh.fm p. 34 Revised; Jnury 29, 2009 Slide 4 Multiplying Out; Methods Σ of Π Π of Σ Three methods of multiplying out Using Using mny times is strightforwrd wy. is very esy to use; unfortuntely, when done utomticlly with no thinking, it cn get very long. If one uses simplifiction (S) X(X+)=X nd bsorption () X(X+)=X t every chnce, the work is shorter, but the finl expression still my be longer thn necessry. Hint: the simplify nd bsorb rules re esier to see in Σ of Π. Tke the dul so one uses X+X=X nd X+X=X+ which re esier to see. Then tke the dul bck to use. Use 2 (or equivlent) nd Swp before using The strnge distributed rule, 2, is the dul of. The equivlent of using 2 is to: tke the dul, use, then reverse the dul. Swp is the sme difficulty in both the dul nd in the originl form, so use it ny time. Following the hint bove, tke the dul nd use (S) nd () t the strt. While in the dul, use. Then reverse the dul nd continue using, remembering to wtch for convenient plces to use Swp, (S), nd (). Using Krnugh mps This is the esiest method for four or five vribles, it lwys gives the smllest nswer, it esily hndles don t cres, but gets very complex for over five input vribles. It is the method of choice for most smll problems. Three methods of fctoring Using 2 This is the strightforwrd wy, unfortuntely it uses the unfmilir 2 distributive lw which mkes the lgebr hrder for most people. Using dulity nd This is lgebriclly equivlent to the previous method. However using the more fmilir () mkes it esier for most people. Using Krnugh mps This is quite esy for four vribles, but more complex for over five input vribles. Esily hndles d s. rleton University dig4fctoringh.fm p. 35, Revised; Jnury 29, 2009 omment on Slide 4

Multiply Out Using st istributive Lw Method I: Using Step : Tke ul, Simplify Step 2: Tke ul bck Step 3: Use Step 2: Simplify, Use Step 3: Simplify, Use... Finl Step: Simplify, Simplify, (K-Mp mt help) Exmple ( + X)( + X)( + X) dul = X + X + X = X(++) f = +X = [( + X) + X( + X)]( + X) = [ + X + X + X]( + X) = [ + X]( + X) = ( + X) + ( + X)X = ( + X + X + X = + X Use () Use () Use (S) x + x = x Use () Use () Use (S) x + cx = x st istributive Lw x( + ) = x + x Wys to Simplify (S) - X +X = X - X +X = X + () - Krnugh mp - onsensus If too mny inputs for mp dig4fctoringh.fm p. 36 Revised; Jnury 29, 2009 Slide 5 Multiply Out Using st istributive Lw Exmple of Multiplying Out Exmple of Multiplying Out Simplifiction, bsorption 2 (or equivlent) nd Swp work well when their re repeted letters in the expression. However if ll the letters re unique, cn only use ( + )( + )(F + G + H) = ( + + + )(F + G + H) = = ( + + + ) F + ( + + + ) G + ( + + + ) H F + F + F + F + G + G + G + G + H + H + H + H ll letters re different, no simplifiction possible Use () rewrite Use () With ll the letters different, there is no wy to simplify. The expressions get long rpidly. Using () lwys works, it is esy on the brin, but hrd on the pencil. lso the simplifictions must be done by other mens. rleton University dig4fctoringh.fm p. 37, Revised; Jnury 29, 2009 omment on Slide 5

Multiply Out Using st istributive Lw Method I: Using Step : Simplify, Use Step 2: Simplify, Use Step 3: Simplify, Use... Finl Step: Simplify Exmple: ( + )( + F + )( + )( + F + ) How to Simplify. 2. X +X = X X +X = X + (S) () 3. Krnugh mp 4. oncensus if you hve to Use twice 0 0 = ( + F + + + F + )( + F + + + F + ) c +cx = c = (F + )(F + + + F + ) bf + bfx = bf = FF + F + F + FF + F Use + F + + + F + ) = F + + + F + ) Simplify using 5-input mp but not done till next chpter NSWER, UT OUL E SIMPLER = F + F + + + n simplify using consensus twice but tricky! F + SKIP THIS TWLE = F + + + = F + ( + + ) ( + ) = F + + consensus consensus st istributive Lw () x( + ) = x + x onsensus F F b + bc + c F = b +c 5-vrible mp F Result is F + + dig4fctoringh.fm p. 38 Revised; Jnury 29, 2009 Slide 6 Multiply Out Using st istributive Lw Exmple of Multiplying Out Note on simplifying by consensus or 5-input mp (continued) The 5-input mp is not covered until the next section. Using the 5-input mp voids the use of consensus in the bove exmple. onsensus is tricky to see. Exmple: Multiply Out using Method I. F is not pure Π of Σ Multiply Out F = [( + )] [ ( + ) + ( + )] ( + + ) Use () nd xx=0 = { [ ( + ) + ( + )] + [ ( + ) + ( + )]} ( + + ) xx = 0 xx = 0 Use () = { [( + )] + [ ( + )]} ( + + ) = = {( + ) + ( + )} ( + + ) = { + + + } ( + + ) Use () nd xx=0 ollect terms; use xy + x = x xx = 0 { + + + } + +{ + + + } = + + + +{ + + + } + + + Plot mp xy + x = x = + + + Mp shows there re no more simplifictions Use () Use () This psudo mtrix helps rrnge the messy multipliction rleton University dig4fctoringh.fm p. 39, Revised; Jnury 29, 2009 omment on Slide 6

Multiply Out; Use 2 (or equ) nd Swp, efore Method II Using 2 nd Swp Step : Simplify (esier to spot in dul) Step 2: Use 2 nd/or Swp, ( & mybe Swp in dul) Step 3: Use, Simplify nd mybe Swp.... Repet Steps 2 nd 3 Exmple: F = ( + + )( + )( + + )( + )( + + ) heck for simplifictions Rerrnge to use (2) nd Swp (x)(+x) = x = ( + + )( + )( + + )( + )( + + ) = ( + + )( + )( + + )( + ) = ( + + )( + + )( + )( + ) 2 Sw F = ( + + ) ( + ) F =( + + ) ( + ) n go no further with 2 nd Swp Must use Go to next slide ommuttive lws 2nd istributive Lw (2) (X+c)(X+d) = X + cd Swp (+X)(b+X)=X+bX Simplifiction (S) Simplifiction(dul) X+X = X (X)(X+) = X Tke dul to use esier forms of (S), () nd F d = ++++ heck for simplifictions F dul = ++++ = +++ Rerrnge to use () nd Swp x+xy = x (S) = +++ Sw = (+)+(+)(+) rcket LL product terms F dul = { (+)}+[(+) (+)] Tke dul bck F = {++( )} [( )+( )] rop extr brkets F = { + + } [ + ] dig4fctoringh.fm p. 40 Revised; Jnury 29, 2009 Slide 7 Multiply Out; Use 2 (or equ) nd Swp, efore Multiplying Out Using 2 nd Swp Multiplying Out Using 2 nd Swp before omments: For the initil step Simplify here mens using the dul rules X(X+)=X nd X(X+)=X However the rules X+X=X nd X+X=X re much esier to pply. pplying 2 here mens (X+)(X+b)=X+b However => X+Xb= X(+b) is esier thn using 2 Swp is self dul nd is eqully obnoxious in both forms. Insted of simplifying s bove, tke the dul s suggested in the brckets, nd s is on the right side of the slide. Tking the dul of Product-of-Sums is esy. fter tht one cn use the esy simplify rules X+X=X nd X+X=X. One cn lso use the simple insted of 2 One cn use swp here or in the next step, it mkes little difference. The next slide multiplies out product terms using => F dul =( + + ) ( + ) This is esier to do in the originl form. Thus tke the dul bck before going on to slide 2. This dul is error prone. The trick is to be sure you put RKETS ROUN LL THE N TERMS! If you think you cn write this on the sme line s converting +, I hve some lnd to sell you!. Using swp while in the dul form introduces more brckets. More brckets increse the error tendncy when tking the reverse dul. Thus there is slight dvntge in using Swp lter. rleton University dig4fctoringh.fm p. 4, Revised; Jnury 29, 2009 omment on Slide 7

Multiply Out Using 2 nd Swp, efore Exmple from lst slide (continued) Originl Problem Multiply Out: F = ( + + )( + )( + + )( + )( + + ) st istributive Lw () x( + ) = x + x On lst slide, using 2 (or equ) nd Swp obtined ( + + ) ( + ) Use () Use () x +xy = x = ( + + ) + ( + + ) ollect terms = + + = + + = + + 0 + + + heck mp for further simplifictions re there ny? dig4fctoringh.fm p. 42 Revised; Jnury 29, 2009 Slide 8 Multiply Out Using 2 nd Swp, efore Exmple With Method II: Multiply out Exmple With Method II: Multiply out using 2 (or equ) nd Sw before using F = ( + )( + + )( + )( + + )( + + ) F d = ( ) + ( ) + ( ) + ( ) + ( ) Tke dul; F d = ( ) + ( ) + ( ) + ( ) + ( ) 0 use +X = F d = ( ) + ( ) + ( ) + ( ) F d = ( + ) + ( + ) F d = [ + ( + )][ + ( + )] F d = [ + + { }] [ + + { }] (dul) 2 =F = [( {+}] + [ ( {+}] Fctor out nd then using Use Swp rcket Ned terms (redy to tke dul) Reverse dul F = + + + Use () This reduces to the sme function s the exmple on Slide 6 The Krnugh mp there shows there re no further simplifictions for the Σ of Π form F = + + + 4-2. PROLEM Multiply out. Remember to check for obvious simplifictions before strting. (W + Y + Z)(X + Y + Z)(W + X)(W + Z)(X + Y + Z)(W + X + Z) Hint: Tke the dul, simplify the dul nd use if possible, then tke the reverse dul, before you multiply out. 4-3. PROLEM Multiply out lgebriclly. The best nswer hs 8 letters. The hint from the lst problem my be useful. ( + )( + + )( + + )( + + )( + + ) rleton University dig4fctoringh.fm p. 43, Revised; Jnury 29, 2009 omment on Slide 8

Multiply Out Using emorgn nd K-Mps Method III: Using K-Mps Step : Find F using Generl emorgn Step 2: Plot F on K-Mp Step 3: Plot F using the 0 Squres Step 4: Get expression for F from mp Exmple: () Find the inverse using generl emorgn F = ( + + )( + )( + + )( + ) F dul = () + ( ) + () + () F = + + + (2) Plot F on K-mp (3) Plot the F on K-mp; Put s where F ws 0 s. (4) ircle F on K-mp F = + + Mp of function, F Mp of function, F Mp of inverse, F dig4fctoringh.fm p. 44 Revised; Jnury 29, 2009 Slide 9 Multiply Out Using emorgn nd K-Mps The Esy Wy to Multiply Out The Esy Wy to Multiply Out F is esy to find using emorgn s lw. F is esy to find from K-mp of F. The bove re the two essentil fcts used for multiplying out using K-mp. When is the K-mp the best method? For most problems done by hnd is by fr the esiest wy to fctor. Six or more vribles will give problem too big for K-mp. Using lgebr my be esier if ll the vribles, or lmost ll, re different.see omment on Slide 5. lgebr my be esier for converting smll prtil expression inside long expression. 4-4. PROLEM: Multiply out using emorgn s Lw nd Krnugh mp, to get two terms of 2 letters nd one of 3 letters. ( + + )( + )( + + )( + )( + + ) 4-5. PROLEM (Solution to lter fctoring problem,nd n erlier lgebric problem) Multiply out to get n expression with eight letters. Use Krnugh mp. ( + )( + + )( + + )( + + )( + + ) 4-6. PROLEM (Remember Simplify, simplify, simplify!) multiply out problem using,,,,e nd F s vribles. ( + )( + + )(E + + )( + + )(E + + )( + F) rleton University dig4fctoringh.fm p. 45, Revised; Jnury 29, 2009 omment on Slide 9

Multiply Out: Using emorgn nd K-Mps Exmple Using Krnugh Mp Steps: () Given F = (Π of Σ expression) F = ( + )( + + )( + + )( + + ) (2) Invert F using emorgn s lw to get F s Σ of Π F = + + + (3) Plot it on mp. (4) Mke mp for F, It hs where F hd 0 (5) ircle the F mp (6) Write out the eqution for F F = + + + Mp of F Mp of F Mp of F dig4fctoringh.fm p. 46 Revised; Jnury 29, 2009 Slide 0 Multiply Out: Using emorgn nd K-Mps Multiply Out With d s Multiply Out With d s If some input combintions re never used, these become don t cre outputs. o the norml steps up through finding F () Mke the mp for F (2) Mke the mp for F from tht of F. Then identify the d s on the mp of F. Finlly circle the mp normlly to find the minimum Σ of Π expression. 4-7. PROLEM: Find the minimum Σ of Π expression for F using the don t cres to best dvntge. F = (W+X+Y)(W+X+Z)(X+Y+Z) The input combintions W XYZ, nd WXYZ never hppen, so these mp squres re d s. Z YZ WX 00 0 0 00 d 0 X W 0 d Y Helpful mp 4-8. PROLEM: Find the minimum Σ of Π expression for G using the don t cres to best dvntge. G = ( + )( + + )( + + )( + + ) The input combintions, nd never hppen. rleton University dig4fctoringh.fm p. 47, Revised; Jnury 29, 2009 omment on Slide 0

Fctoring Fctoring, the ul of Multiplying Out hnge Σ of Π Π of Σ Exmple I ( ) + ( ) + ( ) ( - - - )( - - - )( - - - ) Methods of Fctoring I. Use the 2nd distributive lw x + b = (x + )(x + b) - lwys works, but - very long nd slow. II. Tke the dul Use multiply-out method Tke the dul bck - hnges unfmilir 2 to fmilr - In theory the sme mount of work, but esier to grsp. III. Plot F on K-mp Plot F using the 0 squres Find F using emorgn - Esiest to do. - Gives the simplest Π of Σ nswer. - Very messy for 5 vribles or more. + + Use (2) = (+)(+) + Use (2) = [(+)(+) + ][(+)(+) + ] Use (2) = [(+) + ][(+) + ][(+) + ][(+) + ] Use (2) = [ + ][++ ][++ ][+ + ] nd += = [ + ][+ + ] Use x(x+y) = x Exmple II F = + + F = ( + )+ F = {( + )}+ F UL = {+ )} F UL = + (ul) 2 = F = (+)( + + ) Exmple III F = + Plot on Mp Get F from Mp F = + Use (em) F=( + )( + ) Use () Tke ul Use () Tke ul ck F F dig4fctoringh.fm p. 48 Revised; Jnury 29, 2009 Slide Fctoring Four methods of fctoring Four methods of fctoring I) Using 2 Using 2 mny times is the brute force wy. Unfortuntely students find 2 hrd to use, nd the expnsion my get very long. It helps to use simplifiction X (X+y) = X nd bsorption X (X+y) = Xy t every chnce, but these rules re lso more difficult thn their dul rules. If ll the letters re different, then ll one cn use is 2. II) Using dulity nd This in theory, is just s difficult s the previous method, but the more fmilir rules mkes it seem esier. II) Using bit, then use dulity nd, s in II) bove cn do some fctoring nd often helps t the strt. Exmple I bove, using before 2, cn be done in two lines. ++ =(+) + (using first) ={+}{+(+)} (using 2) III) Using Krnugh mps This is the esiest method for four or five inputs, it lwys gives the smllest nswer, it esily hndles don t cres, but it gets very messy for over five inputs. It is the method of choice for most smll problems. Three methods of multiplying out (compre with fctoring) Using Using mny times is the strightforwrd wy, is very esy to use. Unfortuntely the result cn get very long. Using simplifiction (X + Xy =X) nd bsorption (X + Xy = X y) frequently will help. Using 2 (or equ) nd Swp before using Use 2 or the equivlent (tke the dul nd use ) nd the Swp rule cn do initil consolidtion before using. In most cses one must use for finl clenup. Using Krnugh mps This is quite esy for four vribles, but more complex for over five input vribles. Esily hndles d s rleton University dig4fctoringh.fm p. 49, Revised; Jnury 29, 2009 omment on Slide

Fctoring Using 2 Method III: Using 2 Step : Simplify, Use 2 Repet: Step until done... Unless the problem is very simple the other methods will be esier. 2nd istributive Lw (2) X + cd = (X+c)(X+d) Exmple + X Use (2) ( + X) ( + X) Use (2) gin ( + X)( + X)( + X) Get extended (2) + X = ( + X)( + X)( + X) Exmple + Use (2) gin ( + ) ( + ) Use (2) gin, twice ( + ) ( + )( + ) ( + ) dig4fctoringh.fm p. 50 Revised; Jnury 29, 2009 Slide 2 Fctoring Using 2 Fctoring Using 2 Fctoring Using 2 The expression proven on the slide is: The extended (2) + X = ( + X)( + X)( + X) The dul is the extended () ( + + )X = X + X + X Exmple + ( + ) ( + ) (2) (2) (2) ( + ) ( + )( + ) ( + ) ( + )( + ) ( + ) ( + = ) If we hd strted by using the Swp rule, we would hve simpler nswer in one step. 4-9. PROLEM Fctor + + rleton University dig4fctoringh.fm p. 5, Revised; Jnury 29, 2009 omment on Slide 2

Fctoring Using ulity Method II: Fctoring Using ulity Step. Simplify nd use if possible ut for F dul Step 2. Tke the dul; get fctored, or semi fctored form Step 3. Multiply out the dul to get sum-of products. The right one Step 4. Tke the dul bck to get the fctored form. Fctoring Using ulity The expression to fctor is Σ of Π F = + + () Use F = ( + ) + (2) Tke its dul to get Π of Σ. F = [ ( + { })] + { } rcket LL N terms F UL = [ + ( { + })] { + } (3) Multiply out to get Σ of Π. See box F UL = + + + (4) Tke the dul bck F = ( + )( + )( + )( + ) Get the desired Π of Σ. + + = F = ( + )( + )( + )( + ) Multiply Out etils F UL =[ + { + }] { + } () = [ + + ] { + } = [ + + ] () +[ + + ] bdx+bd=bd F UL = + + + Multiplying out is bsed on (). Esier for people, thn fctoring bsed on (2). lgebr of one is the dul of the lgebr of the other dig4fctoringh.fm p. 52 Revised; Jnury 29, 2009 Slide 3 Fctoring Using ulity hnging Fctoring into Multiplying Out hnging Fctoring into Multiplying Out Fctoring is onverted to Multiplying Out, its ul Problem We tke fctoring problem which is confusing, becuse fctoring is bsed on (2). This lw is not fmilir high-school type lgebric lw nd is hrder to work with. In the dul spce, the dul expression is lredy fctored. The problem is trnsformed into multiplying out, which is bsed on the first distributive lw (). () is more fmilir, nd hence multiplying out is usully esier thn fctoring. Multiplying out in the dul spce does not give the nswer. One tke the dul to get the nswer. This will then be the fctored form of the originl expression. 4-0. PROLEM Show lgebriclly tht F = + + + tkes only 8 letters or 2 gte inputs in fctored form. rleton University dig4fctoringh.fm p. 53, Revised; Jnury 29, 2009 omment on Slide 3

Fctoring Using ulity Method II: Fctoring Using ulity Exmple: equl F = + + + First use () F = ( + ) + ( + ) F =[ ({ } + { })] + [ ({ } + )] Tke dul F dul = [ + ({ + } { + })] [+ ({ + } )] Multiply out = [ + ( + )( + )] [ + ( + )] Sw = ( + )) + ( + )( + ) Sw = ( + )) + ( + ) = + + + First use () twice Put brkets round ll the N terms Minus 25% if you don t differentite between F nd F dul Rerrnge to use Swp Use Swp (+stuff)(+junk)= junk+ stuff Use Swp (+)(+)= + Use () F dul = + + + heck on mp Tke dul bck OK on mp F = ( + + )( + + )( + + )( + + ) Fctored form -> F = ( + + )( + + )( + + )( + + ) mp of F dul dig4fctoringh.fm p. 54 Revised; Jnury 29, 2009 Slide 4 Fctoring Using ulity Method II: Fctoring In the ul Spce Method II: Fctoring In the ul Spce Exmple; Fctor F = + + + F = ( +) + ( + ) F = ( +) + ( + ) F = [+ ( + )] [+( + )] F = [+{ ( + )}] [+({ } + { })] Put brckets round ll Ns redy to tke dul Tke the dul F d = [ {+( )}] + [ ({+} {+})] Remove extr brckets F d = {+( )} + {+} {+} Use () In Generl F d = + + {+} {+} Simplify, use F Use Swp d = + + [ + ] nd mybe Swp fter tking the Use () F d = + + + dul F d = + + + + Mp shows no more simplifictions Tke the dul bck F = ( + ) ( + + )( + + )( + + ) 4-. PROLEM Fctor EF + E + E + EF Use () Use () Use Swp In Generl Simplify, use nd mybe Swp before tking the dul rleton University dig4fctoringh.fm p. 55, Revised; Jnury 29, 2009 omment on Slide 4

Fctoring Krnugh Mps nd emorgn Method III: Fctoring Using K-Mp Step. Plot function F on K-mp Step 2. Plot F by interchnging 0 on the mp. Step 3. ircle the mp to get F. Step 4. Write out the expression for F. Step 5. Use emorgn to get bck F in fctored form. Exmple: Given F = ( Σ of Π expression) F = + + + () Plot it on mp. (2) Mke mp for F, It hs where F hd 0 (3) ircle the F mp (4) Write out the eqution for F F = + + + (5) Invert F using emorgn s lw to get F s Π of Σ F = ( + )( + + )( + + )( + + ) Mp of F Mp of F Mp of F dig4fctoringh.fm p. 56 Revised; Jnury 29, 2009 Slide 5 Fctoring Krnugh Mps nd emorgn Method III: Fctoring Using Krnugh Method III: Fctoring Using Krnugh Mps This method is probbly the esiest, nd lest error prone, for up to four vribles. Five vribles is t lest twice the work of four. bove 5 it gets very messy. It is very esy to incorporte don t cres with this method. 4-2. PROLEM Fctor EF + E + E + EF Using Krnugh mp nd compre your nswer with the previous problem if you did it. 4-3. PROLEM Fctor + + + + Use Krnugh mp nd obtin the minimum Π of Σ expression. 4-4. PROLEM Show, using Krnugh mp, tht F = + + + tkes only 8 letters or 2 gte inputs in fctored form. ompre with Problem 4-0. rleton University dig4fctoringh.fm p. 57, Revised; Jnury 29, 2009 omment on Slide 5

Fctoring Using K-Mps nd emorgn Fctoring Using Krnugh Mps Steps: Given F = ( Σ of Π expression) F = + + + Not the minimum but it doesn t mtter () Plot it on mp. (2) Mke mp for F, It hs where F hd 0 (3) ircle the F mp (4) Write out the eqution for F F = + + (5) Invert F using emorgn s lw to get F s Π of Σ F = ( + )( + + )( + ) Mp of F Mp of F Mp of F dig4fctoringh.fm p. 58 Revised; Jnury 29, 2009 Slide 6 Fctoring Using K-Mps nd emorgn Method III: Fctoring Using Krnugh Exmple: Fctoring 5-Vrible Expression without using mp! Method II: uses initilly, then dulity nd swp before the finl clenup. Tke dul Multiply Out Fctor F = + + + + E Use F = ( + + E) + ( +) F = [ ( +{ } +{ E})] +[ ({ } +)] F dul = [ + ( { + } { + E})] [ + ({ + } )] F dul = [ { + } ] + [ { + } { + E}] F dul = [ { + } ] + [{ + } { + E}] [{ E} + { }] Use ut brckets round LL N terms + Use Swp Redy to use Swp Use Swp F dul = [ { + } ] + [{ E} + { }] F dul = + + E + Use () twice Tke dul bck get (dul) 2 = F = (++)(++)(+++E)(+++) rleton University dig4fctoringh.fm p. 59, Revised; Jnury 29, 2009 omment on Slide 6

Wrnings on t sy (+b+c)(e+b)(+d) is lredy Π of Σ lwys simplify. Look for x+xz = x, x + xy = x + Y, xy + xy = Y before nd fter ech step. Pick the best method: ) for 5 vribles or under use generl emorgn nd mp. ) for 6 vribles or more, use lgebr or find computer progrm. For multiplying out lgebriclly Tke the dul. Simplify nd use Tke the reverse dul. Look for complemented letters, use swp, nd simplify. Use for wht is left. For fctoring lgebriclly EFORE you tke the dul Simplify nd use Tke the dul use Look possible Swp, simplify Use nd simplify Tke the reverse dul Tking the dul The hrd prt is getting the brckets round the LL the N terms. Try: (+ + E ) + ( +) dig4fctoringh.fm p. 60 Revised; Jnury 29, 2009 Slide 7 Originl the slide bove sid, on t sy (+b+c)b(+d) is lredy Π of Σ." ctully it ws Π of Σ, nd the slide embrrsed the lecturer. 4-5. PROLEM Explin why the (+b+c)b(+d) is Π of Σ. Hint: tke the dul. Method III: Fctoring Using Krnugh 4-6. PROLEM ON ULS Tke the dul of F = ( + + E ) + ( + ) To check your nswer look in the omment on Slide 6 I ll bet you cn t get it right the first time without looking. rleton University dig4fctoringh.fm p. 6, Revised; Jnury 29, 2009 omment on Slide 7

dig4fctoringh.fm p. 62 Revised; Jnury 29, 2009 Slide 8 Method III: Fctoring Using Krnugh rleton University dig4fctoringh.fm p. 63, Revised; Jnury 29, 2009 omment on Slide 8