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Exercise Supplement

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Factoring is an essential skill for success in algebra and higher level mathematics courses. Therefore, we have taken great care in developing the student's understanding of the factorization process. The technique is consistently illustrated by displaying an empty set of parentheses and describing the thought process used to discover the terms that are to be placed inside the parentheses. The factoring scheme for special products is presented with both verbal and symbolic descriptions, since not all students can interpret symbolic descriptions alone. Two techniques, the standard "trial and error" method, and the "collect and discard" method (a method similar to the "ac" method), are presented for factoring trinomials with leading coefficients different from 1. This module contains the exercise supplement for the chapter "Factoring Polynomials".

Exercise Supplement

Finding the factors of a Monomial ((Reference))

For the following problems, the first quantity represents the product and the second quantity represents a factor. Find the other factor.

Exercise 1

32 a 4 b, 2b 32 a 4 b, 2b

Solution

16 a 4 16 a 4

Exercise 2

35 x 3 y 2 , 7 x 3 35 x 3 y 2 , 7 x 3

Exercise 3

44 a 2 b 2 c, 11 b 2 44 a 2 b 2 c, 11 b 2

Solution

4 a 2 c 4 a 2 c

Exercise 4

50 m 3 n 5 p 4 q, 10 m 3 q 50 m 3 n 5 p 4 q, 10 m 3 q

Exercise 5

51 (a+1) 2 (b+3) 4 , 3(a+1) 51 (a+1) 2 (b+3) 4 , 3(a+1)

Solution

17( a+1 ) ( b+3 ) 4 17( a+1 ) ( b+3 ) 4

Exercise 6

−26 (x+2y) 3 (x−y) 2 , −13(x−y) −26 (x+2y) 3 (x−y) 2 , −13(x−y)

Exercise 7

−8 x 5 y 4 (x+y) 4 (x+3y) 3 , −2x(x+y) (x+3y) −8 x 5 y 4 (x+y) 4 (x+3y) 3 , −2x(x+y) (x+3y)

Solution

4 x 4 y 4 ( x+y ) 3 ( x+3y ) 2 4 x 4 y 4 ( x+y ) 3 ( x+3y ) 2

Exercise 8

− (6a−5b) 10 (7a−b) 8 (a+3b) 7 , − (6a−5b) 7 (7a−b) 7 (a+3b) 7 − (6a−5b) 10 (7a−b) 8 (a+3b) 7 , − (6a−5b) 7 (7a−b) 7 (a+3b) 7

Exercise 9

12 x n+6 y 2n−5 , −3 x n+1 y n+3 12 x n+6 y 2n−5 , −3 x n+1 y n+3

Solution

−4 x 5 y n−8 −4 x 5 y n−8

Exercise 10

−400 a 3n+10 b n−6 c 4n+7 , 20 a 2n+8 c 2n−1 −400 a 3n+10 b n−6 c 4n+7 , 20 a 2n+8 c 2n−1

Exercise 11

16x−32, 16 16x−32, 16

Solution

( x−2 ) ( x−2 )

Exercise 12

35a−45, 513 35a−45, 513

Exercise 13

24 a 2 −6a, 6a 24 a 2 −6a, 6a

Solution

4a−1 4a−1

Exercise 14

88 x 4 −33 x 3 +44 x 2 +55x, 11x 88 x 4 −33 x 3 +44 x 2 +55x, 11x

Exercise 15

9 y 3 −27 y 2 +36y, −3y 9 y 3 −27 y 2 +36y, −3y

Solution

−3 y 2 +9y−12 −3 y 2 +9y−12

Exercise 16

4 m 6 −16 m 4 +16 m 2 , 4m 4 m 6 −16 m 4 +16 m 2 , 4m

Exercise 17

−5 x 4 y 3 +10 x 3 y 2 −15 x 2 y 2 , −5 x 2 y 2 −5 x 4 y 3 +10 x 3 y 2 −15 x 2 y 2 , −5 x 2 y 2

Solution

x 2 y−2x+3 x 2 y−2x+3

Exercise 18

−21 a 5 b 6 c 4 (a+2) 3 +35 a 5 b c 5 (a+2) 4 , −7 a 4 b (a+2) 2 −21 a 5 b 6 c 4 (a+2) 3 +35 a 5 b c 5 (a+2) 4 , −7 a 4 b (a+2) 2

Exercise 19

−x−2y− c 2 , −1 −x−2y− c 2 , −1

Solution

x+2y+ c 2 x+2y+ c 2

Exercise 20

a+3b, −1 a+3b, −1

Factoring a Monomial from a Polynomial ((Reference)) - The Greatest Common Factor ((Reference))

For the following problems, factor the polynomials.

Exercise 21

8a+4 8a+4

Solution

4( 2a+1 ) 4( 2a+1 )

Exercise 22

10x+10 10x+10

Exercise 23

3 y 2 +27y 3 y 2 +27y

Solution

3y( y+9 ) 3y( y+9 )

Exercise 24

6 a 2 b 2 +18 a 2 6 a 2 b 2 +18 a 2

Exercise 25

21(x+5)+9 21(x+5)+9

Solution

3( 7x+38 ) 3( 7x+38 )

Exercise 26

14(2a+1)+35 14(2a+1)+35

Exercise 27

m a 3 −m m a 3 −m

Solution

m( a 3 −1 ) m( a 3 −1 )

Exercise 28

15 y 3 −24y+24 15 y 3 −24y+24

Exercise 29

r 2 (r+1) 3 −3r (r+1) 2 +r+1 r 2 (r+1) 3 −3r (r+1) 2 +r+1

Solution

( r+1 ) [ r 2 ( r+1 ) 2 −3r( r+1 )+1 ] ( r+1 ) [ r 2 ( r+1 ) 2 −3r( r+1 )+1 ]

Exercise 30

Pa+Pb+Pc Pa+Pb+Pc

Exercise 31

(10-3x)(2+x)+3(10−3x)(7+x) (10-3x)(2+x)+3(10−3x)(7+x)

Solution

( 10−3x )( 23+4x ) ( 10−3x )( 23+4x )

Factoring by Grouping ((Reference))

For the following problems, use the grouping method to factor the polynomials. Some may not be factorable.

Exercise 32

4ax+x+4ay+y 4ax+x+4ay+y

Exercise 33

xy+4x−3y−12 xy+4x−3y−12

Solution

( x−3 )( y+4 ) ( x−3 )( y+4 )

Exercise 34

2ab−8b−3ab−12a 2ab−8b−3ab−12a

Exercise 35

a 2 −7a+ab−7b a 2 −7a+ab−7b

Solution

( a+b )( a−7 ) ( a+b )( a−7 )

Exercise 36

m 2 +5m+nm+5n m 2 +5m+nm+5n

Exercise 37

r 2 +rs−r−s r 2 +rs−r−s

Solution

( r−1 )( r+s ) ( r−1 )( r+s )

Exercise 38

8 a 2 bc+20 a 2 bc+10 a 3 b 3 c+25 a 3 b 3 8 a 2 bc+20 a 2 bc+10 a 3 b 3 c+25 a 3 b 3

Exercise 39

a(a+6)−(a+6)+a(a−4)−(a−4) a(a+6)−(a+6)+a(a−4)−(a−4)

Solution

2( a+1 )( a−1 ) 2( a+1 )( a−1 )

Exercise 40

a(2x+7)−4(2x+7)+a(x−10)−4(x−10) a(2x+7)−4(2x+7)+a(x−10)−4(x−10)

Factoring Two Special Products ((Reference)) - Factoring Trinomials with Leading Coefficient Other Than 1 ((Reference))

For the following problems, factor the polynomials, if possible.

Exercise 41

m 2 −36 m 2 −36

Solution

( m+6 )( m−6 ) ( m+6 )( m−6 )

Exercise 42

r 2 −81 r 2 −81

Exercise 43

a 2 +8a+16 a 2 +8a+16

Solution

( a+4 ) 2 ( a+4 ) 2

Exercise 44

c 2 +10c+25 c 2 +10c+25

Exercise 45

m 2 +m+1 m 2 +m+1

Solution

not factorable

Exercise 46

r 2 −r−6 r 2 −r−6

Exercise 47

a 2 +9a+20 a 2 +9a+20

Solution

( a+5 )( a+4 ) ( a+5 )( a+4 )

Exercise 48

s 2 +9s+18 s 2 +9s+18

Exercise 49

x 2 +14x+40 x 2 +14x+40

Solution

( x+10 )( x+4 ) ( x+10 )( x+4 )

Exercise 50

a 2 −12a+36 a 2 −12a+36

Exercise 51

n 2 −14n+49 n 2 −14n+49

Solution

( n−7 ) 2 ( n−7 ) 2

Exercise 52

a 2 +6a+5 a 2 +6a+5

Exercise 53

a 2 −9a+20 a 2 −9a+20

Solution

( a−5 )( a−4 ) ( a−5 )( a−4 )

Exercise 54

6 x 2 +5x+1 6 x 2 +5x+1

Exercise 55

4 a 2 −9a−9 4 a 2 −9a−9

Solution

( 4a+3 )( a−3 ) ( 4a+3 )( a−3 )

Exercise 56

4 x 2 +7x+3 4 x 2 +7x+3

Exercise 57

42 a 2 +5a−2 42 a 2 +5a−2

Solution

( 6a−1 )( 7a+2 ) ( 6a−1 )( 7a+2 )

Exercise 58

30 y 2 +7y−15 30 y 2 +7y−15

Exercise 59

56 m 2 +26m+6 56 m 2 +26m+6

Solution

2( 28 m 2 +13m+3 ) 2( 28 m 2 +13m+3 )

Exercise 60

27 r 2 −33r−4 27 r 2 −33r−4

Exercise 61

4 x 2 +4xy−3 y 2 4 x 2 +4xy−3 y 2

Solution

( 2x+3y )( 2x−y ) ( 2x+3y )( 2x−y )

Exercise 62

25 a 2 +25ab+6 b 2 25 a 2 +25ab+6 b 2

Exercise 63

2 x 2 +6x−20 2 x 2 +6x−20

Solution

2( x−2 )( x+5 ) 2( x−2 )( x+5 )

Exercise 64

−2 y 2 +4y+48 −2 y 2 +4y+48

Exercise 65

x 3 +3 x 2 −4x x 3 +3 x 2 −4x

Solution

x( x+4 )( x−1 ) x( x+4 )( x−1 )

Exercise 66

3 y 4 −27 y 3 +24 y 2 3 y 4 −27 y 3 +24 y 2

Exercise 67

15 a 2 b 2 −ab−2b 15 a 2 b 2 −ab−2b

Solution

b( 15 a 2 b−a−2 ) b( 15 a 2 b−a−2 )

Exercise 68

4 x 3 −16 x 2 +16x 4 x 3 −16 x 2 +16x

Exercise 69

18 a 2 −6a+ 1 2 18 a 2 −6a+ 1 2

Solution

( 6a−1 )( 3a− 1 2 ) ( 6a−1 )( 3a− 1 2 )

Exercise 70

a 4 +16 a 2 b+16 b 2 a 4 +16 a 2 b+16 b 2

Exercise 71

4 x 2 −12xy+9 y 2 4 x 2 −12xy+9 y 2

Solution

( 2x−3y ) 2 ( 2x−3y ) 2

Exercise 72

49 b 4 −84 b 2 +36 49 b 4 −84 b 2 +36

Exercise 73

r 6 s 8 +6 r 3 s 4 p 2 q 6 +9 p 4 q 12 r 6 s 8 +6 r 3 s 4 p 2 q 6 +9 p 4 q 12

Solution

( r 3 s 4 +3 p 2 q 6 ) 2 ( r 3 s 4 +3 p 2 q 6 ) 2

Exercise 74

a 4 −2 a 2 b−15 b 2 a 4 −2 a 2 b−15 b 2

Exercise 75

81 a 8 b 12 c 10 −25 x 20 y 18 81 a 8 b 12 c 10 −25 x 20 y 18

Solution

( 9 a 4 b 6 c 5 +5 x 10 y 9 )( 9 a 4 b 6 c 5 −5 x 10 y 9 ) ( 9 a 4 b 6 c 5 +5 x 10 y 9 )( 9 a 4 b 6 c 5 −5 x 10 y 9 )

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