Name Date Class _

The Binomial TheoremPractice and Problem Solving: C

+- 5X'1 Lf + 'j 5"

+'1(", 'I.':2.'1~ t lfoX'13 t ';1':1

tox3~;)' t'-fO)(~~3tLOX'J 1ft ~5"

Use the Binomial Theorem to expand each binomial.

1. (x + y)5

X5" t- 5><4ty2. (4x + y)4

~56x ~ t a 5b x "3:;13. (2x + y)5

301 )(5" t 8"0 x '"Ib +4. (n + 2m)4I"yY)if + 3.;:> f)1 30 T of} "".>l." iJ... t 8' "" I) 3 "f- Y1 'f

Solve.

5. At Hopewell High School, 1 in 7 students is on a sports team. Thereare 4 student council representatives in the school.

a. What is the probability that 2 of the student council representativesare also on a sports team?

6,09b. What is the probability that at least 3 of the student council

representatives are on a sports team?

(2,0'6. A donut shop sells donuts with a jelly filling. Two in every 5 donuts

have a jelly filling. There are 5 donuts left in the package.

a. What is the probability that all 5 donuts have a jelly filling?

D. 0 Ib. What is the probability that none of the donuts has a jelly filling?

C> 087. Andrew is choosing CDs from a bag of free CDs without looking. He

has a 1 in 5 chance of choosing a CD that he likes. He chooses 8 CDsin all. What is the probability that he will get 3 CDs that he likes?

6. /58. In a game of bingo, the contestants have a 1 in 12 chance of winning

each round. If Shirley plays 6 rounds, what is the probability that shewill win at least half of them?

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97

Name Date Class

Simplify each polynomial, if possible. Then factor it.

1. 3n' - 48 2. 3x' - 75x

.-. 'Factoring PolynomialsPractice and Problem Solving: AlB

3. 9m' -16

C3vYld. +-'1) t3m~L/J5. 3n6 -12

')( n3 + ••.") ( n3 - ~ ')

7. 3b' + 12b' + 12b

3 b (b3 t;;>.)?'

9. x' - 64

v,-L/) ( X~.f 4x +-Ii»

4. 16r' - 9

(4r;).-t3) (1r;Z-3}

6. x6- 9

( X ~ + ., ) (X 1 - 3 ')

8. 50v6 + 60v' + 18

~(5V3+3)';l-

10. x3 -125

(X-5) (X:J.t5x +025)11. x6 -64 12. x6_1

(X-~)(X:J..r').1C .f/.f ) (xn') ()(~~vrJl) (><-1)(.f.,)( H)(xf'I) (X.2_X .} ~),

Factor each polynomial by grouping.

13. 8n' - 7n' + 56n - 49

{n ;?1/} (ttn-7)15. 9r' +3r' -21r-7

(3(' ;l,_ "1 )( 3r+ I )

17. 120b' + 105b' + 200b + 175

5 (31o~f'5)(8'bt/)

14.5x'-6x'-15x+18

Cx?- 3) ('3x-~)16. 25v' + 25v' -15v -15

J(5V~-3) (v+ ,)18. 120x' - 80x' -168x + 112

8(5x:L-1)(3x -.:t)

Solve.

19. A square concert stage in the center of a fairground has an areaof 4x' + 12x + 9 It'. The dimensions of the stage have the formex + d, where e and d are whole numbers. Find an expression forthe perimeter of the stage. What is the perimeter when x = 2 It?

LJ(;).X +3) ft. a h" £+.)

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101

Name Date _

~.~Factoring PolynomialsPractice and Problem Solving: C

Simplify each polynomial. Then factor it.

1. 12v' -75

3 (~v ~+5)(~VA_5)3. .20u. - 20u3v' + 5v.

. 5 ( d.(/ _ V 3) .;2..

5. 8x3 + 125

(~Xf'~)(~)(J.-IO~ t~5)

Class ------

2. 5p. - 80

?(/ f-4 )(p?> -~ )4. 4x. - 32x3y3 + 64y.

L.j (~ l./ _ (/3 )..2..

6. 64x.-1

(;;''X -I) (Lf X~ t;2x tl)(~)(TI)(L.j)(.<..:c<t1)

Factor each polynomial by grouping. Be sure to factorthe polynomial completely.

7. b3- 2b2

- b + 2 8. 24v' + 56v' -15v _ 35

(b-I)( bt,) (b-~ (~v~5)(3vi7~9. 245n' -175n' -196n + 140

7 (5""n ~-I.f)Dill -5")

11. 150u'v + 75u -125u2 - 90uv

SlA (5/;,-»((011'-5)

10. 140x. + 100x' - 28x' - 20x'

'1X?(5'(.;l-1) ( 7'( +5"~

12. 196x2 Y - 64x + 56x' - 224xy

-.1)((7)(-8) (7'1 +.;t\Solve.

13. Fatima has an herb garden. She grows parsley in a triangular

section having an area of .!.(3x' - 6x - x + 2) tt'. What are the2

dimensions for the base and height of the parsley section?

31( -I f-(~~J X-a ft't'-f

14. The voltage generated by an electrical circuit changes over timeaccording to the polynomial V(t) = t3

- 4t' - 25t + 100, where V is involts and t is in seconds. Factor the polynomial to find the times whenthe voltage is equal to zero. '

(t- 4) (-t-t~/)(iof 5" ') j Th e Va tttJ.tje 15 f-tj k4J +D 2f-~0..+ 'ts CLI\Cl. 55.

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Name Date _

Dividing PolynomialsPractice and Problem Solving: AlB

Class------

Divide by using long division.

1. (x2 -x-6)+(x-3)

3. (_3x2 +20x-12)"i-(x-6)

- 3" +;2

Divide by using synthetic division.

5. (3x2 -8x+4)+(x-2)

7. (9x2 -7x+3)+(x-1)

2. (2x' -10x2 +x-5)+(x-5)

;;'"1 ~ +- I4. (3x' +9x2 -14)+(x+3)

;J. ,<131< -m

6. (5x' -4x+12)+(x+3)

b'j5x-/'i + W

8. (_6x2 +5x-10)+(x+ 7)

_~~ + tj 7 +- 351)H1Use synthetic substitution to evaluate P(x) for the given value.

9. P(x)=4x' -9x+2 for x=3 10. P(x)=-3x2 +10x-4 for x=-2

p (-::I.):: - 3(.,

Determine whether the given binomial is a factor of P(x).

11. (x-4); P(x)=x'+8x-48 12. (x+5); P(x)=2x2-6x-1

-vre~)---- NOSolve.

13. The total number of dollars donated each year to a small charitable organizationhas followed the trend d(t) = 213 + 1012 + 20001 + 10,000, where d is dollars and Iis the number of years since 1990. The total number of donors each year hasfollowed the trend p(l) = 12 + 1000. Write an expression describing the averagenumber of dollars per donor.

dd. t (0

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106

Name . . ..... Dale Cla55 _

•• Dividing Polynomials. Practice and Problem Solving: C

Divide by using long division.

1. (2x' +14x' -4x-48)+(2x+4)

X~+ 51(-l~

2. (x' +12x' -4)+(x-3)

;l. ,31)( t t5)( ""tiS + x:-3

3. (12x' +23x' -9x' +15x+4)+(3x-1)

} J.. '1.~X t-q( rS +- 3x-1

Divide by using synthetic division.

5. (9x' -3x+11)+(x-6)

3'1q( +51 - X-CD

7. (6x5 -3x' +x-2)+(x-1)

'" J:L ~0)(. tk.~ tG,\C +;1' ttf.+;:1

4. (-2x'+11x'-8x-7)+(2x+1)

6. (3x' -2x' +1)+(x+2)

3",3- "I( ~+-tOIC. -~ + ~;l.

8. (-x'-7x'+6x'-1)+(x-3)

3;1. ;17,- X - lOx - .;2.'1)( - 70 -~]

Use synthetic substitution to evaluate PIx) for the given value.

9. P(x)=4x' -12x-2 for x=5 10. P(x)=-3x' +5x' -x+ 7 for x =-2

Use the Factor Theorem to verify that the given binomial is a factorof P(x). Then divide.

11. (x + 5); PIx) = 2x' + 6x - 20 12. (x-1);P(x)=x' -6x' +4x' +1

Solve.

13. The total weight of the cargo entering a seaport each year can bemodeled by the function CIt) = 0.2t' + 1OOOt'+ 1Ot+ 50,000, where tisthe number of years since the port was opened. The average weight ofcargo delivered by each ship is modeled by the functionA(t) = 0.1t + 500. Write an expression describing the number of shipsentering the port each year.

?d)- -t' 10 C

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107