2-3 The Remainder and Factor Theorems

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1 - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x 60 = (x + )(x 5)(x + ) x + x 1x + 18; x So, x + x 1x + 18 = (x )(x + 5x 6) Factorig the quadratic expressio yields x + x 1x + 18 = (x )(x + 6)(x 1) x + x 18x 0; x So, x + x 18x 0 = (x )(x + 7x + 10) Factorig the quadratic expressio yields x + x 18x 0 = (x )(x + )(x + 5) esolutios Maual - Powered by Cogero x + 0x 8x 96; x + Page 1

2 - So, x + x 1x + 18 = (x )(x + 5x 6) The Remaider ad Factor Theorems Factorig the quadratic expressio yields x + x 1x + 18 = (x )(x + 6)(x 1) x + x 18x 0; x So, x + x 18x 0 = (x )(x + 7x + 10) Factorig the quadratic expressio yields x + x 18x 0 = (x )(x + )(x + 5) x + 0x 8x 96; x + So, x + 0x 8x 96 = (x + )(x + 8x ) Factorig the quadratic expressio yields x + 0x 8x 96 = (x + )(x + )(x ) 5 x + 15x + 108x 50; x 6 So, x + 15x + 108x 50 = (x 6)( x x + 90) Factorig the quadratic expressio yields x + 15x + 108x 50 = (x 6)(x + 6)(x 5) 6 6x 7x 9x 1; x + esolutios Maual - Powered by Cogero Page

3 So, x + 15x + 108x 50 = (x 6)( x x + 90) - The Remaider ad Factor Theorems Factorig the quadratic expressio yields x + 15x + 108x 50 = (x 6)(x + 6)(x 5) 6 6x 7x 9x 1; x + So, 6x 7x 9x 1 = (x + )(x 5x ) Factorig the quadratic expressio yields 6x 7x 9x 1 = (x + )(x + 1)(x ) 7 x + 1x + 8x + 1x 6; x + 6x + 9 So, x + 1x + 8x + 1x 6 = (x + 6x + 9)(x + 6x 7) Factorig both quadratic expressios yield x + 1x + 8x + 1x 6 = (x + ) (x + 7)(x 1) 8 x x 6x + 68x + 0; x x 1 So, x x 6x + 68x + 0 = (x x 1)(x + x 0) Factorig both quadratic expressios yield x + 1x + 8x + 1x 6 = (x 6)(x + )(x + 5)(x ) Divide usig log divisio esolutiosmaual- Powered by Cogero 9 (5x x + 6x x + 1) (x ) Page

4 - So, x x 6x + 68x + 0 = (x x 1)(x + x 0) The Remaider ad Factor Theorems Factorig both quadratic expressios yield x + 1x + 8x + 1x 6 = (x 6)(x + )(x + 5)(x ) Divide usig log divisio 9 (5x x + 6x x + 1) (x ) So, = 5x + 17x + 7x (x6 x5 + x x + x x + ) (x + ) So, 5 = x x + 9x 19x + 1x (x 8x + 1x 6x + 1) (x + ) esolutios Maual - Powered by Cogero Page

5 5 - So, The Remaider ad Factor = x Theorems x + 9x 19x + 1x (x 8x + 1x 6x + 1) (x + ) The remaider ca be writte as So, = x 8x + x (x 7x 8x + 10x + 60) (x ) So, = x x 1x 0 1 (6x6 x5 + 6x 15x + x + 10x 6) (x 1) esolutios Maual - Powered by Cogero Page 5

6 - So, The Remaider ad Factor = x Theorems x 1x 0 1 (6x6 x5 + 6x 15x + x + 10x 6) (x 1) = So, 1 (108x5 6x + 75x + 6x + ) (x + ) So, = 6x 6x + x + 9x (x + x + 6x + 18x 16) (x x + 18x 5) esolutios Maual - Powered by Cogero Page 6

7 - So, The Remaider ad Factor = 6x Theorems 6x + x + 9x (x + x + 6x + 18x 16) (x x + 18x 5) So, = x + 16 (x 1x 1x + 110x 8) (x + x 1) So, = x 8x So, = x x esolutios Maual - Powered by Cogero Page 7

8 - So, The Remaider ad Factor Theorems = x x ca be writte as The remaider So, = Divide usig sythetic divisio 19 (x x + x 6x 6) (x ) Because x, c = Set up the sythetic divisio as follows The follow the sythetic divisio procedure The quotiet is x + x + 5x (x + x x + 8x ) (x + ) Because x +, c = Set up the sythetic divisio as follows The follow the sythetic divisio procedure The quotiet is x x + x + 1 (x 9x x 8) (x ) Because x, c = Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i the divided The follow the sythetic divisio procedure esolutios Maual - Powered by Cogero The quotiet is x + x + 1x + + Page 8

9 - The Thequotiet Remaider is x xad + xfactor + Theorems 1 (x 9x x 8) (x ) Because x, c = Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i the divided The follow the sythetic divisio procedure The quotiet is x + x + 1x + + (x5 x + 6x + 9x + 6) (x + ) Because x +, c = Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i the divided The follow the sythetic divisio procedure The quotiet is x x + x + x (1x5 + 10x 18x 1x 8) (x ) Rewrite the divisio expressio so that the divisor is of the form x c Because c = Set up the sythetic divisio as follows, usig a zero placeholder for the missig x-term i the divided The follow the sythetic divisio procedure The remaider ca be writte as So, the quotiet is 6x + 1x + 1x + 1x (6x 6x + 1x 0x 1) (x + 1) Rewrite the divisio expressio so that the divisor is of the form x c esolutios Maual - Powered by Cogero Page 9

10 ca be writte as The remaider So, the quotiet is 6x + 1x + 1x + 1x The Remaider ad Factor Theorems (6x 6x + 1x 0x 1) (x + 1) Rewrite the divisio expressio so that the divisor is of the form x c Set up the sythetic divisio as follows The follow the sythetic divisio procedure Because The quotiet is 1x 6x + 6x 1 5 (5x5 + 6x + x + 8x + 1) (x ) Rewrite the divisio expressio so that the divisor is of the form x c Because c = Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i the divided The follow the sythetic divisio procedure The remaider ca be writte as So, the quotiet is 15x + 1x + 9x + 6x (8x5 + 8x + 68x + 11x + 6) (x + 1) Rewrite the divisio expressio so that the divisor is of the form x c esolutios Maual - Powered by Cogero Page 10

11 ca be writte as The remaider So, the quotiet is 15x + 1x + 9x + 6x + - The Remaider ad Factor Theorems + 6 (8x5 + 8x + 68x + 11x + 6) (x + 1) Rewrite the divisio expressio so that the divisor is of the form x c Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i Because the divided The follow the sythetic divisio procedure ca be writte as The remaider So, the quotiet is 1x + x + 16x x (60x6 + 78x5 + 9x 1x 5x 0) (5x + ) Rewrite the divisio expressio so that the divisor is of the form x c Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i Because the divided The follow the sythetic divisio procedure 5 The quotiet is 1x + 6x x 5 8 esolutios Maual - Powered by Cogero Page 11

12 - The Remaider Theorems 5 ad Factor The quotiet is 1x + 6x x 5 8 Rewrite the divisio expressio so that the divisor is of the form x c Set up the sythetic divisio as follows, usig a zero placeholder for the missig x -term i the Because divided The follow the sythetic divisio procedure 5 The quotiet is 8x 1x + 6x 15 9 EDUCATION The umber of US studets, i thousads, that graduated with a bachelor s degree from 1970 to ca be modeled by g(x) = 0000x 0016x + 051x 715x + 75x , where x is the umber of years sice 1970 Use sythetic substitutio to fid the umber of studets that graduated i 005 Roud to the earest thousad To fid the umber of studets that graduated i 005, use sythetic substitutio to evaluate g(x) for x = 5 The remaider is , so g(5) = Therefore, rouded to the earest thousad,,151,000 studets graduated i SKIING The distace i meters that a perso travels o skis ca be modeled by d(t) = 0t + t, where t is the time i secods Use the Remaider Theorem to fid the distace traveled after 5 secods To fid the distace traveled after 5 secods, use sythetic substitutio to evaluate d(t) for t = 5 The remaider is 50, so d(5) = 50 Therefore, 50 meters were traveled i 5 secods Fid each f (c) usig sythetic substitutio 1 f (x) = x5 x + x 6x + 8x 15; c = esolutios Maual - Powered by Cogero Page 1

13 - The Remaider ad Factor Theorems The remaider is 50, so d(5) = 50 Therefore, 50 meters were traveled i 5 secods Fid each f (c) usig sythetic substitutio 1 f (x) = x5 x + x 6x + 8x 15; c = The remaider is 711 Therefore, f () = 711 f (x) = x6 x5 + x x + 8x ; c = The remaider is 11,165 Therefore, f () = 11,165 f (x) = x6 + 5x5 x + 6x 9x + x ; c = 5 The remaider is 5,56 Therefore, f (5) = 5,56 f (x) = x6 + 8x5 6x 5x + 6x ; c = 6 The remaider is 7,88 Therefore, f (6) = 7,88 5 f (x) = 10x5 + 6x 8x + 7x x + 8; c = 6 The remaider is 67,978 Therefore, f ( 6) = 67,978 6 f (x) = 6x7 + x5 8x + 1x 15x 9x + 6; c = esolutios Maual - Powered by Cogero The remaider is 686 Therefore, f () = f (x) = x8 + 6x5 x + 1x 6x + ; c = Page 1

14 - The Remaider ad Factor Theorems The remaider is 67,978 Therefore, f ( 6) = 67,978 6 f (x) = 6x7 + x5 8x + 1x 15x 9x + 6; c = The remaider is 686 Therefore, f () = f (x) = x8 + 6x5 x + 1x 6x + ; c = The remaider is 15,18 Therefore, f () = 15,18 Use the Factor Theorem to determie if the biomials give are factors of f (x) Use the biomials that are factors to write a factored form of f (x) 8 f (x) = x x 9x + x + 6; (x + ), (x 1) Use sythetic divisio to test each factor, (x + ) ad (x 1) Because the remaider whe f (x) is divided by (x + ) is 0, (x + ) is a factor Test the secod factor, (x 1), with the depressed polyomial x x x + Because the remaider whe the depressed polyomial is divided by (x 1) is 1, (x 1) is ot a factor of f (x) Because (x + ) is a factor of f (x), we ca use the quotiet of f (x) (x + ) to write a factored form of f (x) as f (x) = (x + )(x x x + ) 9 f (x) = x + x 5x + 8x + 1; (x 1), (x + ) Use sythetic divisio to test each factor, (x 1) ad (x + ) Because the remaider whe f (x) is divided by (x 1) is 18, (x 1) is ot a factor Because the remaider whe f (x) is divided by (x + ) is 0, (x + ) is ot a factor 0 f (x) = x x + x + 18x + 15; (x 5), (x + 5) esolutios Maual - Powered by Cogero Use sythetic divisio to test each factor, (x 5) ad (x + 5) Page 1

15 - Because the remaider whe f (x) Because the remaider whe f (x) is is divided by (x 1) isad 18, (xfactor 1) divided by (x + ) is 0, (x + ) is The Remaider Theorems is ot a factor ot a factor 0 f (x) = x x + x + 18x + 15; (x 5), (x + 5) Use sythetic divisio to test each factor, (x 5) ad (x + 5) Because the remaider whe f (x) is divided by (x 5) is 100, (x 5) is ot a factor Because the remaider whe f (x) is divided by (x + 5) is 150, (x + 5) is ot a factor 1 f (x) = x x + 1x + 118x 0; (x 1), (x 5) Use sythetic divisio to test each factor, (x 1) ad (x 5) For (x 1), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure Because the remaider whe f (x) is divided by (x 1) is 0, (x 1) is a factor Test the secod factor, (x 5), with the depressed polyomial x 7x + x + 0 Because the remaider whe the depressed polyomial is divided by (x 5) is 0, (x 5) is a factor of f (x) Because (x 1) ad (x 5) are factors of f (x), we ca use the fial quotiet to write a factored form of f (x) as f (x) = (x 1)(x 5)(x x 8) Factorig the quadratic expressio yields f (x) = (x 1)(x 5)(x )(x + ) f (x) = x x 6x 111x + 0; (x 1), (x 6) Use sythetic divisio to test each factor, (x 1) ad (x 6) For (x 1), rewrite the divisio expressio so that the divisor is of the form x c esolutios Maual - Powered by Cogero Page 15

16 Because the remaider whe the depressed polyomial is divided by (x 5) is 0, (x 5) is a factor of f (x) Because (x 1) ad (x 5) are factors of f (x), we ca use the fial quotiet to write a factored form of f (x) as f - The Remaider ad Factor Theorems (x) = (x 1)(x 5)(x x 8) Factorig the quadratic expressio yields f (x) = (x 1)(x 5)(x )(x + ) f (x) = x x 6x 111x + 0; (x 1), (x 6) Use sythetic divisio to test each factor, (x 1) ad (x 6) For (x 1), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure Because the remaider whe f (x) is divided by (x 1) is 0, (x 1) is a factor Test the secod factor, (x 6), with the depressed polyomial x 9x 0 Because the remaider whe the depressed polyomial is divided by (x 6) is 1, (x 6) is ot a factor of f (x) Because (x 1) is a factor of f (x), we ca use the quotiet of f (x) (x 1) to write a factored form of f (x) as f (x) = (x 1)(x 9x 0) f (x) = x 5x + 8x + 56x + 6; (x ), (x + ) Use sythetic divisio to test each factor, (x ) ad (x + ) For (x ), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure esolutios Maual - Powered by Cogero Because the remaider whe f (x) is divided by (x ) is Page 16, (x ) is ot a factor

17 - Because the remaider whe the depressed polyomial is divided by (x 6) is 1, (x 6) is ot a factor of f (x) Because (x 1) is a factor of f (x), we ca use the quotiet of f (x) (x 1) to write a factored form of f (x) as f The Remaider ad Factor Theorems (x) = (x 1)(x 9x 0) f (x) = x 5x + 8x + 56x + 6; (x ), (x + ) Use sythetic divisio to test each factor, (x ) ad (x + ) For (x ), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure Because the remaider whe f (x) is divided by (x ) is, (x ) is ot a factor Test (x + ) Because the remaider whe f (x) is divided by (x + ) is, (x + ) is ot a factor f (x) = 5x5 + 8x 68x + 59x + 0; (5x ), (x + 8) Use sythetic divisio to test each factor, (5x ) ad (x + 8) For (5x ), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure esolutios Maual - Powered by Cogero Because the remaider whe f (x) is divided by (5x ) is Page 17, (5x ) is ot a factor

18 - The Remaider ad Factor Theorems Because the remaider whe f (x) is divided by (x + ) is, (x + ) is ot a factor f (x) = 5x5 + 8x 68x + 59x + 0; (5x ), (x + 8) Use sythetic divisio to test each factor, (5x ) ad (x + 8) For (5x ), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure Because the remaider whe f (x) is divided by (5x ) is, (5x ) is ot a factor Test (x + 8) Because the remaider whe f (x) is divided by (x + 8) is 1986, (x + 8) is ot a factor 5 f (x) = x5 9x + 9x + x + 75x + 6; (x + ), (x 1) Use sythetic divisio to test each factor, (x + ) ad (x 1) For (x + ), rewrite the divisio expressio so that the divisor is of the form x c Because Set up the sythetic divisio as follows The follow the sythetic divisio procedure + ) is 0, (x + ) is a factor Test the secod factor, (x 1),Page 18 with the depressed polyomial x x + 1x x + 1 esolutios Maualthe - Powered by Cogero Because remaider whe f (x) is divided by (x

19 - The Remaider ad Factor Theorems Because the remaider whe f (x) is divided by (x + 8) is 1986, (x + 8) is ot a factor 5 f (x) = x5 9x + 9x + x + 75x + 6; (x + ), (x 1) Use sythetic divisio to test each factor, (x + ) ad (x 1) For (x + ), rewrite the divisio expressio so that the divisor is of the form x c Set up the sythetic divisio as follows The follow the sythetic divisio procedure Because Because the remaider whe f (x) is divided by (x + ) is 0, (x + ) is a factor Test the secod factor, (x 1), with the depressed polyomial x x + 1x x + 1 Because the remaider whe the depressed polyomial is divided by (x 1) is 8, (x 1) is ot a factor of f (x) Because (x + ) is a factor of f (x), we ca use the quotiet of f (x) (x + ) to write a factored form of f (x) as f (x) = (x + ) (x x + 1x x + 1) 6 TREES The height of a tree i feet at various ages i years is give i the table a Use a graphig calculator to write a quadratic equatio to model the growth of the tree b Use sythetic divisio to evaluate the height of the tree at 15 years a Use the quadratic regressio fuctio o the graphig calculator f(x) = 0001x + x 69 esolutios Maual Powered by of Cogero b To fid -the height the tree at 15 years, use sythetic substitutio to evaluate f (x) for x = 15 Page 19

20 Because the remaider whe the depressed polyomial is divided by (x 1) is 8, (x 1) is ot a factor of f (x) Because (x + ) is a factor of f (x), we ca use the quotiet of f (x) (x + ) to write a factored form of f (x) as f - The Remaider Theorems ad Factor (x) = (x + ) (x x + 1x x + 1) 6 TREES The height of a tree i feet at various ages i years is give i the table a Use a graphig calculator to write a quadratic equatio to model the growth of the tree b Use sythetic divisio to evaluate the height of the tree at 15 years a Use the quadratic regressio fuctio o the graphig calculator f(x) = 0001x + x 69 b To fid the height of the tree at 15 years, use sythetic substitutio to evaluate f (x) for x = 15 The remaider is 985, so f (15) = 985 Therefore, the height of the tree at 15 years is about 985 feet 7 BICYCLING Patrick is cyclig at a iitial speed v0 of meters per secod Whe he rides dowhill, the bike accelerates at a rate a of 0 meter per secod squared The vertical distace from the top of the hill to the bottom of the hill is 5 meters Use d(t) = v0t + at to fid how log it will take Patrick to ride dow the hill, where d(t) is distace traveled ad t is give i secods Substitute v0 =, a = 0, ad d(t) = 5 ito d(t) = v0t + at Use the quadratic equatio to solve for t It will take Patrick 5 secods to travel the 5 meters esolutios Maual - Powered by Cogero Factor each polyomial usig the give factor ad log divisio Assume > 0 8 x + x 1x ; x + Page 0

21 - The Remaider ad Factor Theorems The remaider is 985, so f (15) = 985 Therefore, the height of the tree at 15 years is about 985 feet 7 BICYCLING Patrick is cyclig at a iitial speed v0 of meters per secod Whe he rides dowhill, the bike accelerates at a rate a of 0 meter per secod squared The vertical distace from the top of the hill to the bottom of the hill is 5 meters Use d(t) = v0t + at to fid how log it will take Patrick to ride dow the hill, where d(t) is distace traveled ad t is give i secods Substitute v0 =, a = 0, ad d(t) = 5 ito d(t) = v0t + at Use the quadratic equatio to solve for t It will take Patrick 5 secods to travel the 5 meters Factor each polyomial usig the give factor ad log divisio Assume > 0 8 x + x 1x ; x + So, x + x 1x = (x + )(x x 1) Factorig the quadratic expressio yields x + x 1x = (x + )(x )(x + ) 9 x + x 1x + 10; x 1 esolutios Maual - Powered by Cogero Page 1

22 - So, x + x 1x = (x + )(x x 1) The Remaider Factor Theorems Factorig the quadraticad expressio yields x + x 1x = (x + )(x )(x + ) 9 x + x 1x + 10; x 1 So, x +x 1x + 10 = (x 1)(x + x 10) 50 x + x 10x + ; x + So, x + x 10x + = (x + )(x Factorig the quadratic expressio yields x x + 1) + x 10x + = (x + )(x 1)(x 1) 51 9x + x 171x + 5; x 1 So, 9x + x 171x + 5 = (x 1)(x + 9x 5) Factorig the quadratic expressio yields 9x + x 171x + 5 = (x 1)(x + 6)(x ) 5 MANUFACTURING A 18-ich by 0-ich sheet of cardboard is cut ad folded ito a bakery box esolutios Maual - Powered by Cogero Page

23 - So, 9x + x 171x + 5 = (x 1)(x + 9x 5) Factorig the quadraticad expressio yields 9x + x 171x + 5 = (x 1)(x + 6)(x ) The Remaider Factor Theorems 5 MANUFACTURING A 18-ich by 0-ich sheet of cardboard is cut ad folded ito a bakery box a Write a polyomial fuctio to model the volume of the box b Graph the fuctio c The compay wats the box to have a volume of 196 cubic iches Write a equatio to model this situatio d Fid a positive iteger for x that satisfies the equatio foud i part c a The legth of the box is 18 x The height is x The width of the box is To fid the volume, calculate the product b Evaluate the fuctio for several x-values i its domai The height, width, ad legth of the box must all be positive values For the height, x > 0 For the legth, 18 x > 0 or x < 9 For the width, Thus, the domai of x is 0 < x < > 0 or x < x 0 1 v(x) Use these poits to costruct a graph c Substitute v(x) = 196 ito the origial equatio to arrive at 196 = x 7x +180x d Usig the trace fuctio o a graphig calculator, it appears that v(x) may be 196 i whe x = esolutios Maual - Powered by Cogero Page

24 - Remaider Factor cthe Substitute v(x) = 196ad ito the origialtheorems equatio to arrive at 196 = x 7x +180x d Usig the trace fuctio o a graphig calculator, it appears that v(x) may be 196 i whe x = If x = is a solutio for the equatio, it will also be a solutio to 0 = x 7x + 180x 196 Use sythetic substitutio to verify that x = is a solutio Because the remaider is 0, (x ) is a factor of x 7x + 180x 196 Thus, x = is a solutio to 196 = x 7x +180x Fid the value of k so that each remaider is zero 5 Because x, c = Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is 8 k Solve 8 k = 0 for k Whe k =, will have a remaider of 0 5 Because x +, c = Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is k + 68 Solve k + 68 = 0 for k Whe k =, esolutios Maual - Powered by Cogero 55 will have a remaider of 0 Page

25 k =, - Whe The Remaider ad will have a remaider of 0 Factor Theorems 5 Because x +, c = Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is k + 68 Solve k + 68 = 0 for k Whe k =, will have a remaider of 0 55 Because x + 1, c = 1 Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is k + Solve k + = 0 for k Whe k =, will have a remaider of 0 56 Because x 1, c = 1 Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is k + Solve k + = 0 for k Whe k =, will have a remaider of 0 57 SCULPTING Esteba will use a block of clay that is feet by feet by 5 feet to make a sculpture He wats to reduce the volume of the clay by removig the same amout from the legth, the width, ad the height a Write esolutios Maual Powered by Cogero a -polyomial fuctio to model the situatio Page 5 b Graph the fuctio c He wats to reduce the volume of the clay to of the origial volume Write a equatio to model the situatio

26 Whe k =, will have a remaider of 0 - The Remaider ad Factor Theorems 57 SCULPTING Esteba will use a block of clay that is feet by feet by 5 feet to make a sculpture He wats to reduce the volume of the clay by removig the same amout from the legth, the width, ad the height a Write a polyomial fuctio to model the situatio b Graph the fuctio c He wats to reduce the volume of the clay to of the origial volume Write a equatio to model the situatio d How much should he take from each dimesio? a Let the legth of the block equal feet, the width of the block equal feet, ad the height of the block equal 5 feet Let x equal the amout removed Thus, the legth is ( x), the width is ( x), ad the height is (5 x) The volume of the block is the product of the legth, width, ad height A polyomial fuctio to model the situatio is v(x) = x + 1x 7x + 60 b Evaluate the fuctio for several x-values i its domai The legth, width, ad height of the box must all be positive values For the legth, x > 0 or x < For the width, x > 0 or x < For the width, 5 x > 0 or x < 5 Thus, the domai of x is 0 < x < x v(x) Use these poits to costruct a graph c The volume of the origial block of clay is 5 or 60 cubic feet If Esteba reduces the volume by, the volume of the ew block will be 60 or 6 cubic feet A equatio to model the situatio is 6 = x + 1x 7x + 60 d To solve the equatio 6 = x + 1x 7x + 60 for x, use a graphig calculator to graph y = 6 ad y = x + 1x 7x + 60 o the same scree Use the itersect fuctio from the CALC meu to fid x esolutios Maual - Powered by Cogero Whe x is about 060, v(x) is 6 So, Esteba should take about 060 feet from each dimesio Page 6

27 volume of the ew block will be 60 - or 6 cubic feet A equatio to model the situatio is 6 = x + 1x 7x + 60 The Remaider ad d To solve the equatio 6 =Factor x + 1xTheorems 7x + 60 for x, use a graphig calculator to graph y = 6 ad y = x + 1x 7x + 60 o the same scree Use the itersect fuctio from the CALC meu to fid x Whe x is about 060, v(x) is 6 So, Esteba should take about 060 feet from each dimesio Use the graphs ad sythetic divisio to completely factor each polyomial 58 f (x) = 8x + 6x 10x 156x + 5 The graph suggests that (x + 5) ad (x ) may be factors of the polyomial Use sythetic divisio to test each factor, (x + 5) ad (x ) Because the remaider whe f (x) is divided by (x + 5) is 0, (x + 5) is a factor Test the secod factor, (x ), with the depressed polyomial 8x 1x x + 9 Because the remaider whe the depressed polyomial is divided by (x ) is 0, (x ) is a factor of f (x) Because (x + 5) ad (x ) are factors of f (x), we ca use the fial quotiet to write a factored form of f (x) as f (x) = (x + 5)(x )(8x + 10x ) Factorig the quadratic expressio yields f (x) = (x + 5)(x )(x 1)(x + ) 59 f (x) = 6x5 + 1x 15x + 5x + 7x 80 The graph suggests that (x + 6) ad (x ) may be factors of the polyomial Use sythetic divisio to test each factor, (x + 6) ad (x ) esolutios Maual - Powered by Cogero Page 7 Because the remaider whe f (x) is divided by (x + 6) is 0, (x + 6) is a factor Test the secod factor, (x ), with

28 Because the remaider whe the depressed polyomial is divided by (x ) is 0, (x ) is a factor of f (x) Because (x + 5) ad (x ) are factors of f (x), we ca use the fial quotiet to write a factored form of f (x) as f (x) ad Factor Theorems - =The (x + Remaider 5)(x )(8x + 10x ) Factorig the quadratic expressio yields f (x) = (x + 5)(x )(x 1)(x + ) 59 f (x) = 6x5 + 1x 15x + 5x + 7x 80 The graph suggests that (x + 6) ad (x ) may be factors of the polyomial Use sythetic divisio to test each factor, (x + 6) ad (x ) Because the remaider whe f (x) is divided by (x + 6) is 0, (x + 6) is a factor Test the secod factor, (x ), with the depressed polyomial 6x x 15x + 1x 10 Because the remaider whe the depressed polyomial is divided by (x ) is 0, (x ) is a factor of f (x) The depressed polyomial is 6x 11x 7x + 70 To fid factors of this polyomial, use a graphig calculator to observe the graph The graph suggests that (x ) may be a factor of the depressed polyomial Use sythetic divisio to test the factor (x ) Because the remaider whe the depressed polyomial is divided by (x ) is 0, (x ) is a repeated fact or of f (x) Because (x + 6) ad (x ) are factors of f (x), we ca use the fial quotiet to write a factored form of f (x) as f (x) = (x + 6)(x ) (6x + x 5) Factorig the quadratic expressio yields f (x) = (x + 6)(x ) (x 7)(x + 5) 60 MULTIPLE REPRESENTATIONS I this problem, you will explore the upper ad lower bouds of a fuctio a GRAPHICAL Graph each related polyomial fuctio ad determie the greatest ad least zeros The copy ad complete the table esolutios Maual - Powered by Cogero Page 8 b NUMERICAL Use sythetic divisio to evaluate each fuctio i part a for three iteger values greater tha the greatest zero

29 Because (x + 6) ad (x ) are factors of f (x), we ca use the fial quotiet to write a factored form of f (x) as f (x) = (x + 6)(x ) (6x + x 5) Factorig the quadratic expressio yields f (x) = (x + 6)(x ) (x 7)(x + 5) 60 MULTIPLE REPRESENTATIONS I this problem, you will explore the upper ad lower bouds of a fuctio - The Remaider ad Factor Theorems a GRAPHICAL Graph each related polyomial fuctio ad determie the greatest ad least zeros The copy ad complete the table b NUMERICAL Use sythetic divisio to evaluate each fuctio i part a for three iteger values greater tha the greatest zero c VERBAL Make a cojecture about the characteristics of the last row whe sythetic divisio is used to evaluate a fuctio for a iteger greater tha its greatest zero d NUMERICAL Use sythetic divisio to evaluate each fuctio i part a for three iteger values less tha the least zero e VERBAL Make a cojecture about the characteristics of the last row whe sythetic divisio is used to evaluate a fuctio for a umber less tha its least zero a Use a graphig calculator to graph y = x x 11x + 1 The least zero appears to The greatest zero appears to be Use a graphig calculator to graph y = x + 6x + x 10x The least zero appears to 5 The greatest zero appears to be 1 5 Use a graphig calculator to graph y = x x x The least zero appears to 1 The greatest zero appears to be b Sample aswer: For x x 11x + 1, use sythetic divisio for c = 5, 7, ad 8 esolutios Maual - Powered by Cogero Page 9

30 - The Remaider ad Factor Theorems b Sample aswer: For x x 11x + 1, use sythetic divisio for c = 5, 7, ad 8 f(5) =, f (7) = 180, f (8) = 08 For x + 6x + x 10, use sythetic divisio for c =,, ad f() = 56, f () = 0, f () = 68 5 For x x x, use sythetic divisio for c =, 5, ad 7 f() = 108, f (5) = 50, f (7) = 1,70 c Sample aswer: All of the elemets i the last row of the sythetic divisio are positive d Sample aswer: For x x 11x + 1, use sythetic divisio for c = 5, 7, ad 8 esolutios Maual - Powered by Cogero f( ) = 0, f ( 5) = 108, f ( 6) = 10 Page 0

31 - The Remaider ad Factor Theorems f( ) = 0, f ( 5) = 108, f ( 6) = 10 For x + 6x + x 10, use sythetic divisio for c =,, ad f( 6) = 168, f ( 7) = 560, f ( 9) = 50 5 For x x x, use sythetic divisio for c =, 5, ad 7 f( ) =, f ( ) = 70, f ( ) = 115 e Sample aswer: The elemets i the last row alterate betwee oegative ad opositive 61 CHALLENGE Is (x 1) a factor of 18x165 15x15 + 8x105 15x55 +? Explai your reasoig Yes; sample aswer: Usig the Factor Theorem, (x 1) is a factor if f (1) = 0 Sice f (1) = 0, (x 1) is a factor of the polyomial 6 Writig i Math Explai how you ca use a graphig calculator, sythetic divisio, ad factorig to completely factor a fifth-degree polyomial with ratioal coefficiets, three itegral zeros, ad two o-itegral, ratioal zeros Sample aswer: I would use my graphig calculator to graph the polyomial ad to determie the three itegral zeros, a, b, ad c I would the use sythetic divisio to divide the polyomial by a I would the divide the resultig esolutios Maual - Powered by Cogero Page 1 depressed polyomial by b, ad the the ew depressed polyomial by c The third depressed polyomial will have degree Fially, I would factor the secod-degree polyomial to fid the two o-itegral, ratioal zeros, d ad e So, the polyomial is either the product (x a)(x b)(x c)(x d)(x e) or this product multiplied by some ratioal

32 - The Remaider ad Factor Theorems Sice f (1) = 0, (x 1) is a factor of the polyomial 6 Writig i Math Explai how you ca use a graphig calculator, sythetic divisio, ad factorig to completely factor a fifth-degree polyomial with ratioal coefficiets, three itegral zeros, ad two o-itegral, ratioal zeros Sample aswer: I would use my graphig calculator to graph the polyomial ad to determie the three itegral zeros, a, b, ad c I would the use sythetic divisio to divide the polyomial by a I would the divide the resultig depressed polyomial by b, ad the the ew depressed polyomial by c The third depressed polyomial will have degree Fially, I would factor the secod-degree polyomial to fid the two o-itegral, ratioal zeros, d ad e So, the polyomial is either the product (x a)(x b)(x c)(x d)(x e) or this product multiplied by some ratioal umber 6 REASONING Determie whether the statemet below is true or false Explai If h(y) = (y + )(y + 11y ) 1, the the remaider of is 1 True; sample aswer: The Remaider Theorem states that if h(y) is divided by y ( ), the the remaider is r = h ( ), which is 1 CHALLENGE Fid k so that the quotiet has a 0 remaider 6 Because x + 7, c = 7 Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is 9k 9 Solve 9k 9 = 0 for k Whe k = 9, will have a remaider of 0 65 Because x 1, c = 1 Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is k + 0 Solve k + 0 = 0 for k esolutios Maual Powered by Cogero Whe k = - 0, will have a remaider of 0 Page

33 k = 9, will have a remaider of 0 - Whe The Remaider ad Factor Theorems 65 Because x 1, c = 1 Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is k + 0 Solve k + 0 = 0 for k Whe k = 0, will have a remaider of 0 66 Because x, c = Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is 8k 0 Solve 8k 0 = 0 for k will have a remaider of 0 Whe k = 5, 67 CHALLENGE If x dx + (1 d ) x + 5 has a factor x d, what is the value of d if d is a iteger? x dx + (1 d ) x + 5 ca be writte as x + ( d d + 1)x + 5 Because (x d) is a factor, c = d Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is d + d + 1d + 5 Let d + d + 1d + 5 = 0 ad solve for d Use a graphig calculator to graph y = d + d + 1d + 5 esolutios Maual - Powered by Cogero Page The graph suggests that possible values for d are 5, 0, ad 6 Use sythetic divisio to test each possible factor

34 - Whe The Remaider ad Factor Theorems k = 5, will have a remaider of 0 67 CHALLENGE If x dx + (1 d ) x + 5 has a factor x d, what is the value of d if d is a iteger? x dx + (1 d ) x + 5 ca be writte as x + ( d d + 1)x + 5 Because (x d) is a factor, c = d Set up the sythetic divisio as follows The follow the sythetic divisio procedure The remaider is d + d + 1d + 5 Let d + d + 1d + 5 = 0 ad solve for d Use a graphig calculator to graph y = d + d + 1d + 5 The graph suggests that possible values for d are 5, 0, ad 6 Use sythetic divisio to test each possible factor Sice the remaider is 0, (x ( 5)) or (x + 5) is a factor Test the two remaiig values usig the depressed polyomial Sice the remaider for both is 1, either oe is a factor Thus, d = 5 68 Writig i Math Compare ad cotrast polyomial divisio usig log divisio ad usig sythetic divisio Sample aswer: Both log divisio ad sythetic divisio ca be used to divide a polyomial by a liear factor Log divisio ca also be used to divide a polyomial by a oliear factor I sythetic divisio, oly the coefficiets are used I both log divisio ad sythetic divisio, placeholders are eeded if a power of a variable is missig Determie whether the degree of the polyomial for each graph is eve or odd ad whether its leadig coefficiet a is positive or egative 69 esolutios Maual - Powered by Cogero Page Sice is eve ad a is egative

35 - Sample aswer: Both log divisio ad sythetic divisio ca be used to divide a polyomial by a liear factor Log divisio ca also be used to divide a polyomial by a oliear factor I sythetic divisio, oly the coefficiets are The Remaider ad Factor Theorems used I both log divisio ad sythetic divisio, placeholders are eeded if a power of a variable is missig Determie whether the degree of the polyomial for each graph is eve or odd ad whether its leadig coefficiet a is positive or egative 69 is eve ad a is egative Sice 70 is odd ad a is positive Sice 71 is odd ad a is egative Sice 7 SKYDIVING The approximate time t i secods that it takes a object to fall a distace of d feet is give by Suppose a skydiver falls 11 secods before the parachute opes How far does the skydiver fall durig this time period? Substitute t = 11 ito esolutios Maual - Powered by Cogero ad solve for d Page 5

36 71 - Sice The Remaider ad Factor Theorems is odd ad a is egative 7 SKYDIVING The approximate time t i secods that it takes a object to fall a distace of d feet is give by Suppose a skydiver falls 11 secods before the parachute opes How far does the skydiver fall durig this time period? Substitute t = 11 ito ad solve for d The skydiver falls 196 feet 7 FIRE FIGHTING The velocity v ad maximum height h of water beig pumped ito the air are related by v =, where g is the acceleratio due to gravity ( feet/secod ) a Determie a equatio that will give the maximum height of the water as a fuctio of its velocity b The Mayfield Fire Departmet must purchase a pump that is powerful eough to propel water 80 feet ito the air Will a pump that is advertised to project water with a velocity of 75 feet/secod meet the fire departmet s eeds? Explai a Substitute g = ito v = ad solve for h A equatio that will give the maximum height of the water as a fuctio of its velocity is h = b Substitute v = 75 ito the equatio foud i part a The pump ca propel water to a height of about 88 feet So, the pump will meet the fire departmet s eeds esolutios Maual - Powered by Cogero Solve each system of equatios algebraically 7 5x y = 16 x + y = Page 6

37 - The Remaider ad Factor Theorems The skydiver falls 196 feet 7 FIRE FIGHTING The velocity v ad maximum height h of water beig pumped ito the air are related by v =, where g is the acceleratio due to gravity ( feet/secod ) a Determie a equatio that will give the maximum height of the water as a fuctio of its velocity b The Mayfield Fire Departmet must purchase a pump that is powerful eough to propel water 80 feet ito the air Will a pump that is advertised to project water with a velocity of 75 feet/secod meet the fire departmet s eeds? Explai a Substitute g = ito v = ad solve for h A equatio that will give the maximum height of the water as a fuctio of its velocity is h = b Substitute v = 75 ito the equatio foud i part a The pump ca propel water to a height of about 88 feet So, the pump will meet the fire departmet s eeds Solve each system of equatios algebraically 7 5x y = 16 x + y = 5x y = 16 ca be writte as y = 5x 16 Substitute 5x 16 for y ito the secod equatio ad solve for x Substitute x = ito y = 5x 16 ad solve for y esolutios Maual - Powered by Cogero The solutio to the system of equatio is (, 1) Page 7

38 - The Remaider ad Factor Theorems The pump ca propel water to a height of about 88 feet So, the pump will meet the fire departmet s eeds Solve each system of equatios algebraically 7 5x y = 16 x + y = 5x y = 16 ca be writte as y = 5x 16 Substitute 5x 16 for y ito the secod equatio ad solve for x Substitute x = ito y = 5x 16 ad solve for y The solutio to the system of equatio is (, 1) 75 x 5y = 8 x + y = 1 x + y = 1 ca be writte as x = 1 y Substitute 1 y for x ito the first equatio ad solve for y Substitute y = 1 ito x = 1 y ad solve for x The solutio to the system of equatio is ( 1, 1) 76 y = 6 x x = 5 + y Substitute 6 x for y ito the secod equatio ad solve for x esolutios Maual - Powered by Cogero Page 8

39 - The Remaider ad Factor Theorems The solutio to the system of equatio is ( 1, 1) 76 y = 6 x x = 5 + y Substitute 6 x for y ito the secod equatio ad solve for x Substitute x= 55 ito y = 6 x ad solve for y The solutio to the system of equatio is (55, 075) 77 x + 5y = x + 6y = 5 Elimiate x Solve for y Substitute y = ito the secod equatio ad solve for x The solutio to the system of equatio is 78 7x + 1y = 16 5y x = 1 esolutios Maual - Powered by Cogero 5y x = 1 ca be writte as x + 5y = 1 Elimiate x Page 9

40 - The Thesolutio Remaider adoffactor to the system equatio istheorems 78 7x + 1y = 16 5y x = 1 5y x = 1 ca be writte as x + 5y = 1 Elimiate x Solve for y Substitute y = 1 ito the first equatio ad solve for x The solutio to the system of equatio is (, 1) 79 x + 5y = 8 x 7y = 10 Elimiate x Solve for y Substitute ito the first equatio ad solve for x esolutios Maual - Powered by Cogero The solutio to the system of equatio is Page 0

41 - The Remaider ad Factor Theorems The solutio to the system of equatio is (, 1) 79 x + 5y = 8 x 7y = 10 Elimiate x Solve for y Substitute ito the first equatio ad solve for x The solutio to the system of equatio is 80 SAT/ACT I the figure, a equilateral triagle is draw with a altitude that is also the diameter of the circle If the perimeter of the triagle is 6, what is the circumferece of the circle? A6 B6 C 1 D 1 E 6 If the perimeter of the equilateral triagle is 6, the each side of the triagle measures 1 Also, each agle measures 60 We ca aalyze half of the equilateral triagle, usig the diameter of the circle as oe of the legs esolutios Maual - Powered by Cogero Page 1

42 - The Thesolutio Remaider adoffactor to the system equatio istheorems 80 SAT/ACT I the figure, a equilateral triagle is draw with a altitude that is also the diameter of the circle If the perimeter of the triagle is 6, what is the circumferece of the circle? A6 B6 C 1 D 1 E 6 If the perimeter of the equilateral triagle is 6, the each side of the triagle measures 1 Also, each agle measures 60 We ca aalyze half of the equilateral triagle, usig the diameter of the circle as oe of the legs Sice, this triagle is a right triagle, x = 6 x is also the diameter of the circle The circumferece of a circle is C = d So, the circumferece of the circle is C = The correct aswer is B (6 ) or 6 81 REVIEW If (, 7) is the ceter of a circle ad (8, 5) is o the circle, what is the circumferece of the circle? F 1 G 15 H 18 J 5 K 6 The distace from the ceter of a circle to a poit o the circle is equal to the radius of the circle Use the distace formula ad the two poits to fid the radius of the circle The circumferece of a circle is C = The correct aswer is K d or πr So, the circumferece of the circle is C = (1) or 6 8 REVIEW The first term i a sequece is x Each subsequet term is three less tha twice the precedig term esolutios Maual - Powered by Cogero What is the 5th term i the sequece? A 8x 1 B 8x 15 Page

43 of aad circlefactor is C = Theorems d or πr So, the circumferece of the circle is C = - The Thecircumferece Remaider (1) or 6 The correct aswer is K 8 REVIEW The first term i a sequece is x Each subsequet term is three less tha twice the precedig term What is the 5th term i the sequece? A 8x 1 B 8x 15 C 16x 9 D 16x 5 E x 9 The correct aswer is D 8 Use the graph of the polyomial fuctio Which is ot a factor of x5 + x x x x? F (x ) G (x + ) H (x 1) J (x + 1) The graph suggests that, 1, ad are zeros of the fuctio Thus, (x + ), (x + 1), ad (x ) are factors of f (x) The correct aswer is H esolutios Maual - Powered by Cogero Page

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