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