The Remainder And Factor Theorems
The Remainder and Factor TheoremsYou factoredquadratic expressionsto solve equations.Divide polynomialsusing long divisionand synthetic division.The redwood trees of RedwoodNational Park in California are theoldest living species in the world.The trees can grow up to 350 feetand can live up to 2000 years.Synthetic division can be used todetermine the height of one of thetrees during a particular year.iUse the Remainder!and Factor Theorems.uNewVocabularysynthetic divisiondepressed polynomialsynthetic substitution1 Divide PolynomialsConsider the p o l y n o m i a l function/(x) 6 x - 25x 18* 9. If y o uk n o w that / h a s a zero at x 3, then y o u also k n o w that (x — 3) is a factor o f / ( x ) . Because f(x)is a third-degree polynomial, y o u k n o w that there exists a second-degree polynomial q(x) such that3f(x) 2(X-3)-q(x).This implies that q(x) can be f o u n d by d i v i d i n g 6x — 25x 18x 9 by (x — 3) because32?(*) 7 , i f * 3 .To divide polynomials, we can use an algorithm similar to that of long division w i t h integers.Ise Long Division to Factor PolynomialsFactor 6x — 25x 18x 9 completely using l o n g d i v i s i o n i f (x — 3) is a factor.326x - 7 x - 32-3)6x*-25x 18x 9z( - ) 6* - 18x32-7x(-)Multiply divisor by 6x because y- Gx . -7xSubtract and bring down next term. 21*z22 18x2— —7x.Multiply divisor by —7x becauseSubtract and bring down next term.-3x 9Multiply divisor by - 3 because -3x(-) - 3 * 90-3.Subtract. Notice that the remainder is 0.From this division, y o u can w r i t e 6x — 25x 18x 9 (x — 3)(6x32— 7x — 3).2Factoring the quadratic expression yields 6x — 25x 18* 9 (x — 3)(2x - 3)(3x 1).32So, the zeros of the polynomial function12f{x) 6x - 25x 18* 9 are 3, f, and1——. The x-intercepts of the graph of f(x)32oCAshown support this conclusion.GuidedPractice1-4-2cFactor each p o l y n o m i a l completely using thegiven factor and l o n g d i v i s i o n .IA. x 7x 4x - 12; x 63IB.2\li1!7*f U ) 6 x - 2 5 x 18x 9326x -2x -16x-8;2x-432fl 109
StudyTipProper vs. Improper A rationalexpression is considered improperif the degree of the numerator isgreater than or equal to thedegree of the denominator. So in) Long division of polynomials can result i n a zero remainder, as i n Example 1, or a nonzero\, as i n the example below. Notice that just as w i t h integer long division, the resultof polynomial division is expressed using the quotient, remainder, and divisor.Divisorfix)the division algorithm,x 3Quotient- x 2)x 5x - 4Dividendl(-) x 2xis anRemainder2DividendDivisor -3x-4improper rational expression,whileQuotient(-)3x 6is a proper rationalx 5xx 2lDivisorExcluded value -Remainder — —10expression.Recall that a dividend can be expressed i n terms of the divisor, quotient, and remainder.divisor quotient remainder(x 2) (x 3) (-10) dividend x 5x-42This leads to a definition for polynomial division.Concept Polynomial DivisionLet f(x) and d(x) be polynomials such that the degree of d(x) is less than or equal to the degree of f(x) and d(x) / 0.Then there exist unique polynomials q(x) and r(x) such that ) QW %fo rf[x) d(x).q(x) r(x),where r(x) 0 or the degree of r(x) is less than the degree of d(x). If r(x) 0, then d(x) divides evenly into f{x).K .Before d i v i d i n g , be sure that each polynomial is w r i t t e n i n standard f o r m and that placeholdersw i t h zero coefficients are inserted where needed for missing powers of the variable.StudyTipGraphical Check You can alsocheck the result in Example 2using a graphing calculator. Thegraphs of Yi 9x - x— 3and Y, (3x 2x 1)(3x 2) - 5 are identical.3Example 2 .ong Division with Nonzero RemainderDivide 9x - x - 3 by 3* 2.33 as 9x OxFirst rewrite 9x2c3x 3x 2 ] 9 x 32x l0x -x22-6x3x x2(-) - 6 xYou can w r i t e this result as9r3x 2x l 3x 2'3x 2X( - ) 9x 6x33. Then divide.224x33x TCHECK M u l t i p l y to check this result.3x-3(-) 3x 2[-5, 5] scl: 1 bv [-8, 2] scl: 12x 1 -T(3x 2)(3x - 2x 1) ( - 5 ) 9 x - x - 3239x - 6 x 3x 6 x - 4x 2 - 5 9x - x - 332239x - x - 3 9x - x - 3 33P GuidedPracticeDivide using long division.2A. (8x - 18x 21x - 20) (2x - 3)322B. ( - 3 x3 x 4x - 66) (x - 5)2When d i v i d i n g polynomials, the divisor can have a degree higher than 1. This can sometimes resulti n a quotient w i t h missing terms.110 Lesson 2-3 The Remainder and Factor Theorems
j M j o t HyTipDivision by Polynomial of Degree 2 or HigherD i v i d e 2A - 4A 13x 3x - 11 b y x - 2x 7." n by Zero In Example 3,division is not defined for2x 7 0. From this pointin this lesson, you cane that x cannot take onfor which the indicatedis undefined.43222A'x - 2x 7] 2A - 4A 13x24343A-112( - ) 2A - 4A- 14x32-x 3A - 11l(-) - A 2A - 72A-4You can write this result as2A-4x 13x 3A31121 2A'x- - 2x 72A 7 Guided PracticeD i v i d e using l o n g d i v i s i o n .3A. (2A 5A - 7A 6)32 (A2 3A - 4)3B.(6A - A 12A 15A) (3A - 2A A)54232Synthetic d i v i s i o n is a shortcut for d i v i d i n g a polynomial by a linear factor of the f o r m A — c.Consider the long division from Example 1.v::ce the coefficients highlighted incolored text.Suppress xand powers of x.Collapse the long division vertically,eliminating duplications.Change the signs of the divisor and thenumbers on the second line.6A - 7A - 32-3)6A -25A32( - ) 6A - 18A3 18A 92-3)6 25(-) 6-18-7- 7 A 18A2(-) - 7( - ) - 7 A 21A2-3A 9 1 8 9 2518 2518 18211821 180 21The number now representing thedivisor is the related zero of thebinomial x— c. Also, by changing thesigns on the second line, we are nowadding instead of subtracting.-3 9(-) -3A 9( )-30 90We can use the synthetic division shown i n the example above to outline a procedure for syntheticdivision of any polynomial by a binomial.KeyConcept Synthetic Division AlgorithmTo divide a polynomial by the factor x - c, complete each step.ExampleMM i l l Write the coefficients of the dividend in standard form.Write the related zero c of the divisor x- cm the box.Bring down the first coefficient.Divide 6x - 25x 18x 9 by x - 3.33J6.2- 2 5 - 18,9ETTTiW Multiply thefirst coefficient by c. Write the productunder the second coefficient.Add the product and the second coefficient.EjJiDRepeat Steps 2 and 3 until you reach a sum in the lastcolumn. The numbers along the bottom row are thecoefficients of the quotient. The power of the first termis one less than the degree of the dividend. The finalnumber is the remainder.v„ 'Icoefficientsof quotient Add terms.remainder/ Multiply by c, and writethe product.connectED.mcgraw-hill.com J111
As w i t h division of polynomials b y long division, remember to use zeros as placeholders for anymissing terms i n the d i v i d e n d . When a polynomial is d i v i d e d b y one of its binomial factors x — c,the quotient is called a depressed p o l y n o m i a l .rnthetjc DivisionD i v i d e using synthetic d i v i s i o n .a. (2A - 5x 5x - 2) (x 2)42Because x 2 x — (—2), c —2. Set u p the synthetic division as follows, using zero as aplaceholder for the missing x - t e r m i n the d i v i d e n d . Then follow the syntheticdivision procedure.3 2120Add terms. Multiply by c, andwrite the product.coefficients ofdepressed polynomialTechnologyTipUsing Graphs To check yourdivision, you can graph thepolynomial division expressionand the depressed polynomialwith the remainder. The graphsshould coincide.remainderThe quotient has degree one less than that of the d i v i d e n d , so2x - 5x 5x - 242x2Z (2A -b. (10A - 13A 5A - 14)3Check this result.- 4A 3x - 1.dx 223)Rewrite the division expression so that the divisor is of the f o r m x — c.10x - 13x 5x - 14322x - 35x - -x3(10x - 13x 5A - 14) 2(2x - 3)232 s - 72So, c —. Perform the synthetic division.5- MI-7So, 10x - 13x 5x - 142x-3J- I5A A 4z1325215232or 5x x 42z2x-3Check this result.GuidedPractice4A. (4A 3A - A 8)32 (A -4B. (6A 11A - 15A - 12A 7) (3A 1)3)4322 The Remainder and Factor Theoremsw h e n d(x) is the divisor (A - c) w i t h degree i ,the remainder is the real number r. So, the division algorithm simplifies to/(A) (A - c). q(x) r.Evaluating/(A) for A c, w e f i n d that/(c) (c — c) q(c) r 0 q(c) r or r.So,/(c) r, w h i c h is the remainder. This leads us to the f o l l o w i n g theorem.KeyConcept Remainder TheoremIf a polynomial f{x) is divided by x— c, the remainder is r f(c).112I Lesson 2-3 The Remainder and Factor Theorems
M1 HiThe Remainder Theorem indicates that to evaluate a polynomial function/(x) for x c, y o u cand i v i d e / ( x ) by x — c using synthetic division. The remainder w i l l be/(c). Using synthetic divisionto evaluate a function is called synthetic substitution.Real-World ExampleUse the Remainder TheoremFOOTBALL The n u m b e r of tickets sold d u r i n g the Northside H i g h School f o o t b a l l season canbe modeled b y t(x) x — 12x 48x 74, where x is the n u m b e r of games played. Use theRemainder Theorem to f i n d the n u m b e r of tickets sold d u r i n g the t w e l f t h game of theNorthside H i g h School f o o t b a l l season.32To f i n d the number of tickets sold d u r i n g the twelfth game, use synthetic substitution toevaluate t(x) for x 12. eal-WorldLink to" school football rules ares r arto most college andirrrssional football rules. TwoTiacr differences are that theuaters are 12 minutes aszccsed to 15 minutes and kicking 3ke place at the 40-yard line"S" d of the 30-yard line.Sarce: National Federation of StateHigh School Associations12J1-1248740576120The remainder is 650, so (12) 650.Therefore, 650 tickets were sold d u r i n gthe twelfth game of the season.48 650CHECK You can check your answer using direct substitution.t(x)12x2 48x 74Original function(12) - 12(12) 48(12) 74 or 650 t(12)3Substitute 12 for x and simplify.2 Guided Practice5. FOOTBALL Use the Remainder Theorem to determine the number of tickets sold d u r i n gthe thirteenth game of the season.If y o u use the Remainder Theorem to evaluate/(x) at x c and the result is/(c) 0, then y o u k n o wthat c is a zero of the function and (x — c) is a factor. This leads us to another useful theorem thatprovides a test to determine whether (x — c) is a factor o f / ( x ) .Concept Factor TheoremA polynomial f[x) has a factor (x - c) if and only if f(c) 0.You can use synthetic division to p e r f o r m this test.j.t»,iiiijt ; dUse the Factor TheoremUse the Factor Theorem to determine i f the b i n o m i a l s g i v e n are factors of fix).b i n o m i a l s that are factors to w r i t e a factored f o r m of fix).Use thea. fix) 4x 21x 25x - 5x 3; (x - 1), ix 3)432Use synthetic division to test each factor, (x — 1) and (x 3).il442125-53425504525504548Because the remainder w h e n / ( x ) isd i v i d e d by (x - 1) is 4 8 , / ( l ) 48 and(x — 1) is not a factor.-3J442125-5-12-2769-213-3Because the remainder w h e n / ( x ) isdivided by (x 3) is 0 , / ( - 3 ) 0 and(x 3) is a factor.Because (x 3) is a factor of fix), we can use the quotient of / ( x ) -f (x 3) to write afactored f o r m o f / ( x ) .fix) (x 3)(4x 9 x - 2x 1)320
TechnologyTipCHECK If (x 3) is a factor o f / ( x ) 4 x 21x 4325x — 5x 3, then —3 is a zero of the functionand (—3, 0) is an x-intercept of the graph.Graph/(x) using a graphing calculator andconfirm that (—3, 0) is a point on the graph.Zeros You can confirm the zeroson the graph of a function byusing the zero feature on theCALC menu of a graphingcalculator. 10,10] scl: 1 by [-10, 30] scl: 2b. fix) 2 x - x - 41x - 20; ix 4), ix - 5)3zUse synthetic division to test the factor (x 4).-4 2- 12-41-20-83620-9-50Because the remainder w h e n / ( x ) is divided by (x 4) is 0,/(—4) 0 and (x 4) is afactor o f / ( x ) .Next, test the second factor, (x - 5), w i t h the depressed polynomial 2x — 9x - 5.25J2-9-5105Because the remainder w h e n the quotient o f / ( x ) -f (x 4) is d i v i d e d by (x - 5) is 0,/(5) 0 and (x — 5) is a factor o f / ( x ) .Because (x 4) and (x — 5) are factors o f / ( x ) , we can use the final quotient to write a factoredform o f / ( x ) .fix) ix 4)(x - 5)(2x 1)CHECK The graph o f / ( x ) 2 x - x - 41x - 2032confirms that x - 4 , x 5, and x - arezeros of the function.-10,10] scl: 1 by [-120, 80] scl: 20w GuidedPracticeUse the Factor Theorem to determine i f the b i n o m i a l s given are factors of fix).b i n o m i a l s that are factors to w r i t e a factored f o r m of fix).6A. fix) 3 x3x - 22x 24; (x - 2), (x 5)6B. fix) 4 x334x 54x 36; (x - 6), (x - 3)Use the22You can see that synthetic division is a useful tool for factoring and finding the zeros of polynomialfunctions.ConceptSummary Synthetic Division and RemaindersIf r is the remainder obtained after a synthetic division of f{x) by (x - c), then the following statements are true. ris the value of f(c). If r 0, then (x - c) is a factor of f{x). If r 0, then c is an x-intercept of the graph of f. If r 0, then x c is a solution of f(x) 0. 114 Lesson 2-3 The Remainder and Factor Theorems
Exercises Step-by-Step Solutions begin on page R29.r actor each p o l y n o m i a l completely using the given factorand l o n g d i v i s i o n . (Example 1)1. x 2x - 23x - 60; x 43230. SKIING The distance i n meters that a person travels o nskis can be modeled b y d(t) 0.2f 3t, where t is thetime i n seconds. Use the Remainder Theorem to f i n d thedistance traveled after 45 seconds. (Example 5)22. x 2 x - 2 1 x 1 8 ; x - 332F i n d each/(c) using synthetic substitution. (Example 5)3. x 3x - 18x - 40; x - 43231. fix) 4 x - 3 x x - 6x 8x - 15; c 354. 4 x 20x - 8 x - 9 6 ; x 3326.62254333. fix) 2 x 5 x - 3 x 6 x - 9 x 3x - 4; c 56x - Ix - 29x - 12; 3x 43332. f{x) 3 x - 2 x 4x - 2 x 8x - 3; c 45. - 3 x 15x 108x - 540; x - 6346254327. x 12x 38x 12* - 63; x 6x 934. f{x) 4 x 8x - 6 x - 5 x 6x - 4; c 6a x - 3 x - 36x 68x 240; x - 4x - 1235. fix) 10x 6x - 8 x 7 x - 3x 8; c -6434236225210.543211. (4x - 8 x 12x - 6x 12)435854342545342243( x x 6 x 18x - 216)ia(4x - 14x - 14x HOx - 84)„6x - 12x 10x - 2x - 8x 83x 2x 332-r240. fix) x - 2 x 24x 18x 135; (x - 5), (x 5)( x - 3 x 18x - 54)3234«.23343(2x x - 12)-r22242. / ( x ) 4 x - x - 36x - l l l x 30; (4x - 1), (x - 6)43243. fix) 3 x - 35x 38x 56x 64; (3x - 2), (x 2)43333212x 5x - 15x 19x - 4x - 283x 2x - x 64241. / ( x ) 3 x - 22x 13x 118x - 40; (3x - 1), (x - 5)445339. fix) x 2 x - 5 x 8x 12; (x - 1), (x 3)14 (108x - 36x 75x 36x 24) (3x 2)5438. fix) x - 2 x - 9 x x 6; (x 2), (x - 1)213. (6x - 3 x 6x - 15x 2 x lOx - 6) (2x - 1)42Use the Factor Theorem to determine i f the b i n o m i a l s givenare factors of fix). Use the b i n o m i a l s that are factors to w r i t ea factored f o r m of fix). (Example 6)(x 2)(2x 4)-r236337. f(x) - 2 x 6 x - 4 x 12x - 6x 24; c 412. (2x - 7 x - 38x 103x 60) -r (x - 3)422( x - 2 x x - x 3 x - x 24)6479. (5x - 3 x 6 x - x 12) (x - 4)3336. fix) - 6 x 4 x - 8x 12x - 15x - 9x 64; c 2Divide using l o n g d i v i s i o n . (Examples 2 and 3)453244. f{x) 5 x 38x - 68x 59x 30; (5x - 2), (x 8)2542245. / ( x ) 4 x - 9x 39x 24x 75x 63; (4x 3), (x - 1)5432ide using synthetic d i v i s i o n . (Example 4)46. TREES The height of a tree i n feet at various ages i n yearsis given i n the table.«L( x - x 3 x - 6x - 6)&(2x 4 x - 2 x 8x - 4) -i- (x 3)A(3x - 9 x - 24x - 48) (x - 4)432434(x - 2)-r2322. ( x - 3 x 6 x 9x 6) (x 2)521at32(12x 10x - 18x - 12x - 8) (2x - 3)5432(36x - 6 x 12x - 30x - 12) - j - (3x 1)43225. (45x 6 x 3 x 8x 12)5SL4(3x - 2)34327. (60x 78x 9 x - 12x - 25x - 20) (5x 4)6—54354322& EDUCATION The number of U.S. students, i n thousands,that graduated w i t h a bachelor's degree from 1970 to 2006can be modeled b y g(x) 0.0002x - 0.016x 0.512x .15x 47.52x 800.27, where x is the number of yearssince 1970. Use synthetic substitution to f i n d the numberof students that graduated i n 2005. Round to the nearestI thousand. (Example 0.736111.5b. Use synthetic division to evaluate the height of the treeat 15 years.16x - 56x - 24x 96x - 42x - 30x 1052x-763.3a. Use a graphing calculator to write a quadraticequation to model the g r o w t h of the tree.(48x 28x 68x l l x 6) (4x 1)52347. BICYCLING Patrick is cycling at an initial speed v of4 meters per second. When he rides d o w n h i l l , the bikeaccelerates at a rate a of 0.4 meter per second squared.The vertical distance from the top of the h i l l to the bottom1 9of the h i l l is 25 meters. Use d(t) v t -ar to f i n d h o wlong it w i l l take Patrick to ride d o w n the h i l l , where d(t) isdistance traveled and t is given i n seconds.QQconnectED.mcgraw-hill.com 115
Factor each p o l y n o m i a l u s i n g the g i v e n factor and l o n gd i v i s i o n . Assume n 0.48. x " xln- 14x - 24; x" 249. x " x2n- 12x 10; x" - 13360.jLE REPRESENTATIONS In this problem, y o u w i l lexplore the upper and lower bounds of a function.a. GRAPHICAL Graph each related polynomial function,and determine the greatest and least zeros. Then copyand complete the table.nn50. 4 x " 2x " - lOx" 4; 2x 4351. 9x23nn2nGreatestZeroPolynomial 24x " - 171x 54; 3x - 1n3x52. MANUFACTURING A n 18-inch by 20-inch sheet of cardboardis cut and folded into a bakery box.2x211x 1LeastZero2x 6x 3 x - 1 0 x432x - x - 2x543b. NUMERICAL Use synthetic division to evaluate eachfunction i n part a for three integer values greater thanthe greatest zero.18 in.i!C. VERBAL Make a conjecture about the characteristics ofthe last row w h e n synthetic division is used toevaluate a function for an integer greater than itsgreatest zero.20 in.a. Write a polynomial function to model the volumeof the box.d. NUMERICAL Use synthetic division to evaluate eachfunction i n part a for three integer values less than theleast zero.b. Graph the function.C. The company wants the box to have a volume of196 cubic inches. Write an equation to model thissituation.e. VERBAL Make a conjecture about the characteristicsof the last row w h e n synthetic division is used toevaluate a function for a number less than its leastzero.d. Find a positive integer for x that satisfies the equationfound i n part c.F i n d the value of k so that each remainder is zero.54. x 18x kx 4x 2for 2x53.3x- 2H.O.T. Problems56. 2x - x x kx- 155. x 4x — kx 1x 132322Use Higher-Order Thinking Skills(fi) CHALLENGE Is (x - 1) a factor of 1 8 x15x 4? Explain your reasoning.165- 15x135 8x105-5557. SCULPTING Esteban w i l l use a block of clay that is 3 feetby 4 feet by 5 feet to make a sculpture. He wants toreduce the volume of the clay by removing the sameamount f r o m the length, the w i d t h , and the height.a. Write a polynomial function to model the situation.62. WRITING IN MATH Explain h o w y o u can use a graphingcalculator, synthetic division, and factoring to completelyfactor a fifth-degree polynomial w i t h rational coefficients,three integral zeros, and t w o non-integral, rational zeros.b. Graph the function.C. He wants to reduce the volume of the clay to — ofthe original volume. Write an equation to modelthe situation.63. REASONING Determine whether the statement below istrue or false. Explain.If h{y) iy 2)(3y l l y - 4) - 1, then the remainder2d. H o w much should he take from each dimension?Use the graphs and synthetic d i v i s i o n to completely factoreach p o l y n o m i a l .58. f{x) 8x 26x - 103x - 156x 45 (Figure 2.3.1)43259. fix) 6x 13x - 153x 54x 724x - 840(Figure 2.3.2)543ofCHALLENGE Find k so that the quotient has a 0 remainder.64.x for - 34x 56x 765.x fa: - 8x 173x - 16x - 120x- 166.kx 2x - 22x - 4x-2I'tOUUB200T\wo-8-4 \-1600I 1116322\/iRnn yFigure 2.3.1is - 1 .y 21/ M/fix)64332267. CHALLENGE I f 2x - dx (31 - d )x 5 has a factorw h a t is the value of d i f d is an integer?28xFigure 2.3.2 Lesson 2-3 The Remainder and Factor Theorems2x-d,68. WRITING IN MATH Compare and contrast polynomialdivision using long division and using synthetic division.
Spiral ReviewDetermine whether the degree n of the p o l y n o m i a l for each graph is even or odd and whetherits leading coefficient a is positive or negative. (Lesson 2-2)n70.I I I I /!j m71.72. SKYDIVING The approximate time t i n seconds that i t takes an object to fall a distance ofd feet is given by t Suppose a skydiver falls 11 seconds before the parachute opens.H o w far does the skydiver fall d u r i n g this time period? (Lesson 2-1)73. FIRE FIGHTING The velocity v and m a x i m u m height h of water being p u m p e d into the air arerelated by v \J2gh, where g is the acceleration due to gravity (32 feet/second ). (Lesson 1-7)2a. Determine an equation that w i l l give the m a x i m u m height of the water as a function ofits velocity.b. The M a y f i e l d Fire Department must purchase a p u m p that is p o w e r f u l enough to propelwater 80 feet into the air. W i l l a p u m p that is advertised to project water w i t h a velocityof 75 feet/second meet the fire department's needs? Explain.Solve each system of equations algebraically. (Lesson 0-5)74. 5x165x-y- v 162x 3y 375. 3x-5y 77. 2x 5y 43x 6y 578. 7x 12y 165y - 4x - 2 176. y 6 — xx 4.5 yx 2y 179. 4x by - 83x - 7y 10Skills Review for Standardized Tests80. SAT/ACT I n the figure, an equilateral triangle is d r a w nw i t h an altitude that is also the diameter of the circle.If the perimeter of the triangle is 36, w h a t is thecircumference of the circle?82. REVIEW The first term i n a sequence is x. Eachsubsequent term is three less than twice the precedingterm. What is the 5th term i n the sequence?A 8x — 21C 16x-39B 8x - 15D 16x-45E 32x - 4383. Use the graph of the p o l y n o m i a l function. W h i c h isnot a factor of x x - 3 x - 3 x - 4x - 4?5A 6V27TC 12V2TTB 6V37TD12V37TG15TTH 18TTJ 25TT32F (x - 2 )E 36TTyG (x 2)H81. REVIEW If (3, - 7 ) is the center of a circle and (8,5) ison the circle, what is the circumference of the circle?F 13TT4-4!(x - 1 )' —HJ (x 1)-o12K 26-TT\\\{V/ / fix)f -3x -3x -4x-4432
So,/(c) r, which is the remainder. This leads us to the following theorem. KeyConcept Remainder Theorem If a polynomial f{x) is divided by x— c, the remainder is r f(c). 112 I Lesson
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In Example 2, the remainder is 0. Therefore, x 21 is a factor of x3 x 2. If f(x) 3x3 x2 2, then f( 1) ( 1) ( 1) 2 2 or 0. This illustrates the Factor Theorem, which is a special case of the Remainder Theorem. Use the Remainder Theorem to find the remainder when 2 x 3 3 x 2 x is divided by x 1. State whe
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Euclidean Algorithm (p. 102) To find gcd(a, b) where b a: Divide b into a and let r 1 be the remainder. Divide r 1 into b and let r 2 be the remainder. Divide r 2 into r 1 and let r 3 be the remainder. Continue to divide the remainder into the divisor until you get a remainder of zero. gcd(a, b) the last nonzero remainder.
