Extra Support for Factoring Worksheet

 

By: Dilshad Akrayee

Georgia Highlands College

Rome, GA 30161

 

 

Notes

 

1)     To factor a polynomial completely, look for:

a.      greatest common factor, GCF

b.      a perfect square trinomial

c.      a different of two squares

d.      a pair of binomial factors

e.      a grouping of terms

2)     Factor it out if there is a common factor.

3)     If it is binomial (two terms), check if it is the difference of two squares.

4)     If it is trinomial (three terms), check if it is a perfect square trinomial. If is not a perfect square, check if the coefficient of squared term is 1 then use the method in section 5.2. If the coefficient of the squared terms is not equal to 1, use general factoring methods.

5)     If the polynomial has more than three terms, try to factor by using the grouping of terms.

6)     Check that each factor is prime.

7)     Check your answer by multiplying all the factors.

 

 

The following exercises include various kinds of factoring methods:

 

1. 32A9 + 16A5 + 24A3

2. 2A2 10A 48

3. 14M3 + 7M2 70M

4. 9X2 + 64

5. 6X2 + 31X + 5

6. A2 3AB 4B2

7. 49X2 16Y2

8. 100B2C2 + 30BC3 50B2C

9. 16X + 20

10.     2A2 + 5A 3

11.     10Y2 - 7YX 6X2

12.     Y4 16

13.     A2 + 2A 15

14.     6Y2 5Y 4

15.     32X3 +56X2 16X

16.     15Y + 5

17.     X2 12X + 36

18.     9A2 64

19.     Y2 4YM 12M2

20.     16X2 8X + 1

21.     6Y2 6Y 12

22.     72Y3X2 + 12Y2 24Y4X2

23.     E2 17E + 66

24.     A2 + 17A + 72

25.     M2 + 9

26.     108A2 36A + 3

27.     X2 3XA 10A2

28.     45A3B5 60A4B2 + 75A5B4

29.     4M2 12M + 9

30.     A2 3AB 28B2

31.     16C2 + 24CA + 9A2

32.     3M2 + 4M 4

33.     3M3 12M2 15M

34.     A4 625

35.     16M2 48M + 36

36.     8M2 10M 3

37.     36Y6 42Y5 120Y4

38.     8E2 + 23E 3

39.     5X3 45X2 + 70X

40.     8M2 2MN 3N2

41.     54A2 24X2

42.     4M2 20MX + 25X2

43.     6A2 +10A 4

44.     15N2 + 11NE 14E2

45.     A2 81

46.     10X2 7X 6

47.     125A4 400A3B + 195A2B2

48.     9Y2 + 12Y 5

49.     A2 4A + 4

50.     27E10 45E9 252E8

51.     24M4E + 60M3E2 + 125M2E3

52.     10A2 +25A 60

53.     12E2 + EQ 6Q2

54.     M2 64

55.     64E2 100A2

56.     2A2 + 7AB 15B2

57.     100A2 81Y2

58.     8A2 + 23AB 3B2

59.     A2 + 8A + 16

60.     4Y2 25

61.     X2 = 64

62.     2X+4

63.     6 X2 3x 4x + 2

64.     X2 + 3X + 2

65.     -

66.     - 8

67.     +

68.     +8

69.     X2 + 2X + 1

70.     X2 - 2X + 1


Answers:

1. 32A9 + 16A5 + 24A3 = 8A3(4A6 + 2A2 + 3)

2. 2A2 10A 48 = 2(A2 5A 24)= 2 (A 8)(A + 3)

3. 14M3 + 7M2 70M = 7M(2M + 5)(M 2)

4. 9X2 + 64 Prime

5. 6X2 + 31X + 5 = (X + 5 )6X + 1)

6. A2 3AB 4B2 = (A 4B)(A + B)

7. 49X2 16Y2 = (7X + 4Y)(7X 4Y)

8. 100B2C2 + 30BC3 50B2C = 10BC(10BC + 3C2 5B)

9. 16X + 20 = 4(4X + 5)

10.     2A2 + 5A 3 = (A + 3)(2A 1)

11.     10Y2 - 7YX 6X2 = (5Y 6X)(2X + Y)

12.     Y4 16 = (Y2 + 4)(Y + 2)(Y 2)

13.     A2 + 2A 15 = (A + 5)(A 3)

14.     6Y2 5Y 4 = (3X 4)(2X + 1)

15.     32X3 + 56X2 16X = 8X(X + 2)(4X 1)

16.     15Y + 5 = 5(3Y + 1)

17.     X2 12X + 36 = (X 6)(X 6) = (X 6)2

18.     9A2 64 = (3A + 8)(3A 8)

19.     Y2 4YM 12M2 = (Y 6M)(Y + 2M)

20.     16X2 8X + 1 = (4X - 1)(4X 1) = (4X 1)2

21.     6Y2 6Y 12 = 6(Y 2)(Y + 1)

22.     72Y3X2 + 12Y2 24Y4X2 = 12Y2(6YX2 + 1 2Y2X2)

23.     E2 17E + 66 = (E 11)(E 6)

24.     A2 + 17A + 72 = (A + 8)(A + 9)

25.     M2 + 9 Prime

26.     108A2 36A + 3 = 3(36A2 12A + 1)

=3(6A 1)(6A 1)

27.     X2 3XA 10A2 = (X 5A)(X + 2A)

28.     45A3B5 60A4B2 + 75A5B4

= 15A3B2 (3B3 4A + 5A2B2)

29.     4M2 12M + 9 = (2M 3)2

30.     A2 3AB 28B2 = (A - 7B)(A + 4B)

31.     16C2 + 24CA + 9A2 = (4C + 3A)2

32.     3M2 + 4M 4 = (M + 2)(3M 2)

33.     3M3 12M2 15M = 3M(M 5)(M + 1)

34.     A4 625 =(A2 + 25)(A + 5)(A 5)

35.     16M2 48M + 36 = (4M 6)2

36.     8M2 10M 3 = (2M 3)(4M + 1)

37.     36Y6 42Y5 120Y4 = 6Y4 (6Y2 7Y 20)

=6Y4 (2Y 5)(3Y + 4)

38.     8E2 + 23E 3 = (E + 3)(8E 1)

39.     5X3 45X2 + 70X = 5X(X2 9X + 14)

= 5X(X 7) (X 2)

40.     8M2 2MN 3N2 = (4M 3N)(2M + N)

41.     54A2 24X2 = 6(9A2 4X2)

= 6(3A +2X) (3A 2X)

42.     4M2 20 MX + 25X2 = (M 5X)2

43.     6A2 + 10A 4 = 2(3A2 + 5A 2)

=2(3A 1) (A + 2)

44.     15N2 + 11NE 14E2 = (5N + 7)(3N 2)

45.     A2 81 = (A + 9)(A 9)

46.     10X2 7X 6 = (2X + 1)(5X 6)

47.     125A4 400A3B + 195A2B2

= 5A2 (25A2 80AB + 39B2)

=5A2 (5A 13B) (5A 3B)

48.     9Y2 + 12Y 5 = (3Y + 5)(3Y 1)

49.     A2 4A + 4 = (A 2)(A 2) = (A 2)2

50.     27E10 45E9 252E8 = 9E8(3E2 5E 28)

= 9E8 (3E + 7) (E 1)

51.     24M4E + 60M3E2 + 125M2E3 =

= M2E (24M2 + 60ME + 125E2)

52.     10A2 + 25A 60 = 5(2A2 + 5A 12)

= 5(A + 4) (2A 3)

53.     12E2 + EQ 6Q2 = (4E + 3)(3E 2)

54.     M2 64 = (M + 8)(M 8)

55.     64E2 100A2 = 4(16E2 25A2)

= 4(4E + 5A) (4E 5A)

56.     2A2 + 7AB 15B2 = (2A 3B)(A + 5B)

57.     100A2 81Y2 = (10A + 9Y)(10A 9Y)

58.     8A2 + 23AB 3B2 = (8A B)(A + 3B)

59.     A2 + 8A + 16 = (A + 4)(A + 4) = (A + 4 )2

60.     4Y2 25 = (2Y + 5)(2Y 5)

61.     X2 = 64

X2 64 = (X-8) (X+8)

62.     2X+4 = 2 (X+2)

63.     6 X2 3x 4x + 2

= (6 X2 3x) + ( 4x + 2)

= 3x (2x 1) + -2(2x - 1)

= (2x 1) (3x 2)

64.     X2 + 3X + 2 = (X+1)(X+2)

65.     - = (A B) (A2 + AB + B2 )

66.     - 8 = - = (A 2) (A2 + 2A + 4 )

67.     + = (A + B) (A2 - AB + B2 )

68.     + 8 = + = (A + 2) (A2 - 2A + 4 )

69.     X2 + 2X + 1 = (X+1)(X+1) = (X+1)2

70.     X2 - 2X + 1 = (X-1)(X-1) = (X-1)2