What GOAL 1 - ClassZone

1 Page 1 of 2 The remainder and FactorTheoremsDIVIDINGPOLYNOMIALSWhen you divide a polynomial (x) by a divisor d(x), you get a quotient polynomialq(x) and a remainder polynomial r(x). We write this as d((xx)) =q(x)+ dr((xx)) . Thedegree of the remainder must be less than the degree of the divisor. Example 1 shows how to divide polynomials using a method called Using Polynomial Long DivisionDivide 2x4+ 3x3+ 5x 1 by x2 2x+ division in the same format you would use when dividing numbers. Include a 0 as the coefficient of x2. 2xx24 7xx23 1x02x2 2x2+ 17x+10x2 2x+2 2 x4 + 3 x3 + 1 0 x2 + 1 5 x 1 1 2x4 4x3+ 14x2 Subtract2x2(x2 2x+2).7x3 14x2+ 15x7x3 14x2+ 14xSubtract7x(x2 2x+2).10x2 19x 1110x2 20x+ 20 Subtract10(x2 2x+2).11x 21remainderWrite the result as follows. = 2x2+ 7x+ 10+ CHECK You can check the result of a division problem by multiplying the divisorby the quotient and adding the remainder .

2 The result should be the dividend.(2x2+ 7x+ 10)(x2 2x+2)+ 11x 21= 2x2(x2 2x+2) + 7x(x2 2x+2)+ 10(x2 2x+2) + 11x 21= 2x4 4x3+ 4x2+7x3 14x2+ 14x+ 10x2 20x + 20 + 11x 21= 2x4+ 3x3+ 5x 1 11x 21 x2 2x+22x4+3x3+5x 1 x2 2x+2 EXAMPLE 1long 6 Polynomials and Polynomial FunctionsDivide polynomialsand relate the result to theremainder theorem and thefactor polynomialdivision in real-lifeproblems,such as finding a productionlevel that yields a certainprofit in Example 5. To combine two real-lifemodels into one new model,such as a model for moneyspent at the movies each year in Ex. you should learn itGOAL2 GOAL1 what you should each stage, divide the term withthe highest power in what s left ofthe dividend by the first term of thedivisor. This gives the next term ofthe 1 of remainder and Factor Theorems353In the activity you may have discovered that (2) gives you the remainder when (x)is divided by x 2.

3 This result is generalized in the remainder may also have discovered in the activity that synthetic substitution gives thecoefficients of the quotient. For this reason, synthetic substitution is sometimes calledIt can be used to divide a polynomial by an expression of theform x Synthetic DivisionDivide x3+ 2x2 6x 9 by (a) x 2 and (b) x+ synthetic division for k= 2. = x2+ 4x+ 2 + x 52 find the value of k, rewrite the divisor in the form x k. Because x+3=x ( 3), k= 3. = x2 x 3x3+ 2x2 6x 9 x+ 3x3+ 2x2 6x 9 x 2 EXAMPLE 2synthetic a polynomial (x) is divided by x k, then the remainder is r= (k). remainder THEOREMSTUDENTHELPS tudy TipNotice that syntheticdivision could nothavebeen used to divide thepolynomials in Example 1because the divisor,x2 2x+2, is not of theform x 1 2 6 9284142 5 3 1 2 6 9 3 3 91 1 3 0 Investigating Polynomial DivisionLet (x) = 3x3 2x2+ 2x long division to divide (x) by x 2.

4 what is the quotient? what is theremainder?Use synthetic substitution to evaluate (2). How is (2) related to theremainder? what do you notice about the other constants in the last row ofthe synthetic substitution?21 DevelopingConceptsACTIVITYPage 1 of 2354 Chapter 6 Polynomials and Polynomial FunctionsIn part (b) of Example 2, the remainder is 0. Therefore, you can rewrite the result as:x3+ 2x2 6x 9 = (x2 x 3)(x+ 3)This shows that x+ 3 is a factor of the original from Chapter 5 that the number kis called a zero of the function because (k)= a PolynomialFactor (x) = 2x3+ 11x2+ 18x+ 9 given that ( 3) = ( 3) = 0, you know that x ( 3) or x+ 3 is a factor of (x). Use synthetic division to find the other result gives the coefficients of the + 11x2+ 18x+ 9 = (x+ 3)(2x2+ 5x+ 3)= (x+ 3)(2x+ 3)(x+ 1)Finding Zeros of a Polynomial FunctionOne zero of (x)=x3 2x2 9x+ 18 is x = 2. Find the other zeros of the find the zeros of the function, factor (x) completely.

5 Because (2) = 0, you knowthat x 2 is a factor of (x). Use synthetic division to find the other result gives the coefficients of the quotient. (x) = (x 2)(x2 9)Write (x) as a product of two (x 2)(x+ 3)(x 3)Factor difference of squares. By the factor theorem, the zeros of are 2, 3, and 4 EXAMPLE 3A polynomial (x) has a factor x kif and only if (k) = THEOREM 3 2 11 18 9 6 15 925 302 1 2 9 1820 1810 9 0 HOMEWORK HELPV isit our Web extra 1 of remainder and Factor Theorems355 USINGPOLYNOMIALDIVISION INREALLIFEIn business and economics, a function that gives the price per unit pof an item interms of the number xof units sold is called a demand function. Using Polynomial ModelsACCOUNTINGYou are an accountant for a manufacturer of radios. The demandfunction for the radios is p=40 4x2where xis the number of radios produced inmillions. It costs the company $15 to make a an equation giving profit as a function of the number of radios company currently produces million radios and makes a profit of$24,000,000, but you would like to scale back production.

6 what lesser number ofradios could the company produce to yield the same profit? 24for Pin the function you wrote in part (a).24= 4x3+25x0= 4x3+25x 24 You know that x= is one solution of the equation. This implies that x a factor. So divide to obtain the following: 2(x )(2x2+3x 8)=0 Use the quadratic formula to find thatx is the other positive solution. The company can make the sameprofit by selling 1,390,000 units. CHECK Graph the profit function toconfirm that there are two productionlevels that produce a profit of$24,000, 5 GOAL2 Profit=Revenue Cost= Profit = (millions of dollars)Price per unit =(dollars per unit)Number of units =(millions of units)Cost per unit =15(dollars per unit)= 15P= 4x3+25xxx(40 4x2)Px40 4x2 PNumberof unitsCostper unitNumberof unitsPriceper unitProfitLABELSVERBALMODELALGEBRAICMODE LN umber of units (millions) (millions of dollars) ProductionPROBLEMSOLVINGSTRATEGYACCOUNTA NTMost people think ofaccountants as working formany clients.

7 However, it iscommon for an accountantto work for a single client,such as a company or ONCAREERSPage 1 of 2356 Chapter 6 Polynomials and Polynomial the remainder a polynomial division problem that you would use long division to write a polynomial division problem that you would use synthetic divisionto the polynomial divisor, dividend, and quotient represented by thesynthetic division shown at the using polynomial long (2x3 7x2 17x 3) (2x+3)5.(x3+5x2 2) (x+4)6.( 3x3+4x 1) (x 1)7.( x3+2x2 2x+3) (x2 1)Divide using synthetic (x3 8x+ 3) (x+3)9.(x4 16x2+ x+ 4) (x+4)10.(x2+ 2x+ 15) (x 3)11.(x2+ 7x 2) (x 2)Given one zero of the polynomial function, find the other (x)=x3 8x2+4x+ 48; 413. (x)=2x3 14x2 56x 40; back at Example 5. If the company produces 1 millionradios, it will make a profit of $21,000,000. Find another number of radios thatthe company could produce to make the same using polynomial long division.

8 15.(x2+ 7x 5) (x 2)16.(3x2+ 11x+ 1) (x 3)17.(2x2+ 3x 1) (x+ 4)18.(x2 6x+ 4) (x+ 1)19.(x2+ 5x 3) (x 10)20.(x3 3x2+ x 8) (x 1)21.(2x4+ 7) (x2 1)22.(x3+ 8x2 3x+ 16) (x2+ 5)23.(6x2+ x 7) (2x+ 3)24.(10x3+ 27x2+ 14x+ 5) (x2+ 2x)25.(5x4+ 14x3+ 9x) (x2+ 3x)26.(2x4+ 2x3 10x 9) (x3+ x2 5)USINGSYNTHETICDIVISIOND ivide using synthetic division. 27.(x3 7x 6) (x 2)28.(x3 14x+ 8) (x+ 4)29.(4x2+ 5x 4) (x+ 1)30.(x2 4x+ 3) (x 2)31.(2x2+ 7x+ 8) (x 2)32.(3x2 10x) (x 6)33.(x2+ 10) (x+ 4)34.(x2+3) (x+3)35.(10x4+ 5x3+ 4x2 9) (x+ 1)36.(x4 6x3 40x+ 33) (x 7)37.(2x4 6x3+ x2 3x 3) (x 3)38.(4x4+ 5x3+ 2x2 1) (x+ 1)PRACTICEANDAPPLICATIONSGUIDEDPRACTICEV ocabulary Check Concept Check Skill Check STUDENTHELPHOMEWORK HELPE xample 1:Exs. 15 26 Example 2:Exs. 27 38 Example 3:Exs. 39 46 Example 4:Exs. 47 54 Example 5:Exs. 60 62 STUDENTHELPE xtra Practiceto help you masterskills is on p.

9 948. 3 1 2 9 18 3 15 181 5 6 0 Page 1 of remainder and Factor Theorems357 FACTORINGF actor the polynomial given that (k)= (x) = x3 5x2 2x+ 24; k= 240. (x) = x3 3x2 16x 12; k=641. (x) = x3 12x2+ 12x+ 80; k=1042. (x) = x3 18x2+ 95x 126; k=943. (x) = x3 x2 21x+ 45; k= 544. (x) = x3 11x2+ 14x+ 80; k=845. (x) = 4x3 4x2 9x+9; k=146. (x) = 2x3+7x2 33x 18; k= 6 FINDINGZEROSG iven one zero of the polynomial function, find the other (x) = 9x3+ 10x2 17x 2; 248. (x) = x3+ 11x2 150x 1512; 1449. (x) = 2x3+ 3x2 39x 20; 450. (x) = 15x3 119x2 10x+ 16; 851. (x) = x3 14x2+47x 18; 952. (x) = 4x3+ 9x2 52x+ 15; 553. (x) = x3+ x2+2x+ 24; 354. (x) = 5x3 27x2 17x 6; 6 You are given an expression for the volume of therectangular prism. Find an expression for the missing 3x3+ 8x2 45x 2x3+ 17x2+ 40x+ 25 POINTS OFINTERSECTIONFind all points of intersection of the two graphsgiven that one intersection occurs at x= divide two polynomials and obtain the result5x2 13x+47 x1+022.

10 what is the dividend? How did you find it? demand function for a type of camera is given by the model p= 100 8x2where pis measured in dollars per camera and xismeasured in millions of cameras. The production cost is $25 per camera. Theproduction of million cameras yielded a profit of $ million. what othernumber of cameras could the company sell to make the same profit? 1980 to 1991, the total fuel consumption T(in billions of gallons) by cars in the United States and the average fuelconsumption A(in gallons per car) can be modeled byT= + + 72and A= + 580where xis the number of years since 1980. Find a function for the number of carsfrom 1980 to 1991. About how many cars were there in 1990?104xyy x3 6x2 6x 3y x2 7x 263xyy x3 x2 5xy x2 4x 2x 5x 1?x 5x 1?GEOMETRYCONNECTIONALTERNATIVEFUELJ oshua and Kaia Tickell builtthe Green Grease Machine,which converts usedrestaurant vegetable oil intobiodiesel fuel.