Factoring Review - manhassetschools.org€¦ · Factoring Review In Grade 10, you learned many ways of factoring: 1. GCF factoring 2. Difference of Squares factoring 3. Simple trinomial

Factoring Review

In Grade 10, you learned many ways of factoring:

1. GCF factoring

2. Difference of Squares factoring

3. Simple trinomial factoring

4. Complex trinomial factoring

This is a review of these methods of factoring to aid us in the next topic of sinusoidal functions, rational

expressions and trigonemtirc identities.

GCF Factoring

The greatest common factor of two numbers (x and y) is defined as the largest factor that can be evenly

divided into each number (x and y).

Example 1: Find the GCF of the numbers 20 and 50.

Solution: The factors of 20 are 1, 2, 4, 5, 10, 20.

Because 1 x 20 = 20; 2 x 10 = 20; 4 x 5 = 20

The factors of 50 are 1, 2, 5, 10, 25, 50.

Because 1 x 50 = 50; 2 x 25 = 50; 5 x 10 = 50

Therefore, 10 is the largest factor that is common to both 20 and 50,

therefore it is the GCF.

Example 2: Find the GCF of the terms 9x10y8z9 and 12x9y8z10 and 18x8y9

Solution: The factors of 9x10y8z9 are 1, 3, 9, x10, y8, z9

The factors of 12x9y8z10 are 1, 2, 3, 4, 6, 12, x9, y8, z10

The factors of 18x8y9 are 1, 2, 3, 6, 9, 18, x8, y9

Therefore, the GCF is 3x8y8.

GCF factoring is a method where you find the GCF of all your terms and factor it out of the brackets

(divide each term by the GCF).

Example 3: Factor cba 61215

Solution:

Example 4: Factor xxx 23

Solution:

Example 5: Factor 326443 yxyxyx

Solution:

Example 6: Factor 236 885664 ccc

Solution:

Example 7: Factor 5005 x

Solution:

)245(3

3

6

3

12

3

153

cba

cba

)1( 2

23

xxx

x

x

x

x

x

xx

)1( 3232

32

32

32

64

32

4332

yxxyyx

yx

yx

yx

yx

yx

yxyx

)1178(8

8

88

8

56

8

648

42

2

2

2

3

2

62

ccc

c

c

c

c

c

cc

)100(5

5

500

5

55

x

x

Factoring: greatest common factor (gcf)

Factor the following expressions by using the greatest common factor.

1) –5x3 + 6x

2) 12x3 + 4x2 – 14x

3) -2x3 + 22x

4) -312x3 - 234x2 + 260x

5) 96x3 - 192x2 + 24x

6) –175x2 + 75x

7) -189x2 + 105x

8) 24x3 + 21x

9) -6x3 - 9x

10) 150x4 - 175x2

11) 9x2 + 60

12) x2 – 7x4 + x6

13) z6 + z2

14) 6x3 + x2

15) 270x3 – 300x2 – 30x

16) –42x3 - 336

17) 5x5 + 2x4 + 10x3

18) –4x2 + 12x

19) –18x3 + 36x2 – 90x

20) 3x4 – 16

21) 9a2 - 18a 22) 16a5b3 + 32a4b

23) x2 + x4 + x3 24) 3x5 + 4x4 - 5x2

25) 2x3 - x 26) 3a5 - a3

27) 32b2 + 16b 28) 5x3 – 7x2

29) 3x2 – 10x3 30) a5n + a3n

31) x3 – 5x2 32) 9c – 3c2

33) 5x4 - 12x2 34) x2 + x

35) 6x2 – 12x3 – 18x4 36) x3y4 + x2y2

37) 18b – 9b2 38) 2x3 + 6x2

39) 12x3 + 4x2 40) x5 + 3x2

Answers:

1. )65( 2 xx

2. )726(2 2 xxx

3. )11(2 2 xx

4. )10912(26 2 xxx

5. )184(24 2 xxx

6. )37(25 xx

7. )3563(3 xx

8. )78(3 2 xx

9. )32(3 2 xx

10. )76(25 22 xx

11. )203(3 2 x

12. )71( 422 xxx

13. )1( 42 zz

14. )16(2 xx

15. )1109(30 2 xxx

16. )8(42 3 x

17. )1025( 23 xxx

18. )3(4 xx

19. )52(18 2 xxx

20. 163 4 x

21. )2(9 aa

22. )2(16 24 abba

23. )1( 22 xxx

24. )543( 232 xxx

25. )12( 2 xx

26. )13( 23 aa

27. )12(16 bb

28. )75(2 xx

29. )103(2 xx

30. )1( 23 nn aa

31. )5(2 xx

32. )3(3 cc

33. )125( 22 xx

34. )1( xx

35. )321(6 22 xxx

36. )1( 222 xyyx

37. )2(9 bb

38. )3(2 2 xx

39. )13(4 2 xx

40. )3( 32 xx

Difference of Squares Factoring (DOS)

The difference of squares method can only be used if your expression/equation is a binomial and both

terms are perfect squares subtracted from each other, i.e. 492 x .

When using this method you will end up with two brackets, one with a subtraction sign and one with an

addition sign, and the square roots of the terms in each.


Solution: 77 xx because

749

2

xx

Example 2: Factor 4)( 2 xy

Solution: 22 xyxy because

24

2

xyxy

Example 3: Factor 100222 hnm

Solution: 1010 mnhmnh because

10100

222

mnhhnm


Solution: cannot use DOS factoring because 3x is not a perfect square


Solution: 66 33 xx because

636

36

xx

Example 6: Factor 281 y

Solution: yy 99 because

yy

2

981

Example 7: Factor 2516 4 x

Solution: 5454 22 xx because

525

416 24

xx

FACTORING: Difference of squares

Factor the following expressions by using DOS.

1) 4x2 – 9

2) 25x2 – 121

3) x2 – 25

4) x2 – 1

5) 25 - x2

6) x2 – 100

7) 2x2 – 9

8) 196x2 – 1

9) 4a2 – 121

10) 9y2 – 16

11) 1 – 100x2

12) 36 – 194c2

13) 0.01x2 – 16

14) 16x2 + 9

15) 162 x

16) 492 y

17) 14 2 x

18) 481 2 x

19) 12116 2 x

20) 3649 2 x

21) 291 x

22) 28116 x

23) 10022 yx

24) 2522 yx

25) 42 x

26) 2225 yx

27) 2264 yx

28) 224 yx

29) 42 1649 yx

30) 12 a

31) 162 c

32) 362 a

33) 92 b

34) 812 y

Answers:

1. 3232 xx

2. 115115 xx

3. 55 xx

4. 11 xx

5. xx 55

6. 1010 xx

7. can’t factor by DOS

8. 114114 xx

9. 112112 aa

10. 4343 yy

11. xx 101101

12. can’t factor using DOS

13. 41.041.0 xx

14. can’t factor using DOS

15. 44 xx

16. 77 yy

17. 1212 xx

18. 2929 xx

19. 114114 xx

20. 6767 xx

21. xx 3131

22. xx 9494

23. 1010 xyxy

24. 55 xyxy

25. 22 xx

26. xyxy 55

27. xyxy 88

28. yxyx 22

29. 22 4747 yxyx

30. 11 aa

31. 44 cc

32. 66 aa

33. 33 bb

34. 99 yy

Simple Trinomial Factoring

This method of factoring can only be used if you have trinomial expression where the leading coefficient

is equal to 1. i.e. 1682 xx .

When using this method you need two find two numbers that multiply to give you the ending coefficient

and add to give you the middle coefficient. Those two numbers are placed in two brackets with the

square root of the first term. It may be useful to set up a diagram to organize your thoughts.

Example 1: Factor 1682 xx

Solution: +16 therefore, 44 xx

Example 2: Factor 1272 zz

Solution: +12 therefore, 34 zz

Example 3: Factor 322 aa

Solution: -3 therefore, 13 aa

Example 4: Factor 262 xx

Solution: cannot factor by simple trinomial method, there are no two numbers that

multiply to give you -2 and add to give you +6.

+4 +4 = +8

-4 -3 = -7

-3 +1 = -2

Factoring: simple trinomials

Factor the following expressions.

1) x2 + 10x + 21

2) x2 + 3x + 2

3) x2 - 6x + 8

4) x2 - 5x - 14

5) x2 - 5x + 4

6) y2 – 10y + 16

7) x2 - 11x + 30

8) d2 + 20d + 80

9) x2 + 15x + 54

10) a2 + 12a + 35

11) b2 – 13b + 42

12) x2 + 15x + 26

13) a2 – 4a – 21

14) x2 + 18x – 77

15) x2 - x - 56

16) g2 + 9g + 14

17) x2 + x – 110

18) x2 + 11x + 24

19) x2 + 4x + 12 20) x2 + 4x – 21

21) y2 – 8y + 12

22) x2 + 8x - 15

23) x2 - x - 20

24) x2 – x – 72

25) 542 xx

26) 50152 xx

27) 3242 xx

28) 672 xx

29) 11122 xx

30) 20122 xx

31) 3522 xx

32) 72182 xx

33) 56152 xx

34) 1662 xx

35) 1582 xx

36) 722 xx

37) 39162 xx

38) 121222 xx

39) 12132 xx

40) 22 23 yxyx

41) 22 2414 yxyx 42) 22 65 yxyx

43) 22 632 yxyx

44) 22 338 yxyx

45) 1682 xx 46) 20122 xx

47) 11122 xx 48) 202 cc

49) 36122 xx 50) 62 xx

51) 35122 xx 52) 1892 xx

53) 42132 yy 54) 4062 xx

55) 1322 xx 56) 22 338 yxyx

57) 22 2410 baba 58) 22 23 nmnm

59) 22 4415 yxyx 60) 42232 tt

61) 36122 yy 62) 4542 bb

63) 1832 nn 64) 21102 cc

ANSWERS:

1. (x + 3)(x + 7)

2. (x + 2)(x + 1)

3. (x – 4)(x – 2)

4. (x – 7)(x + 2)

5. (x – 4)(x – 1)

6. (y – 2) (y – 8)

7. (x – 5)(x – 6)

8. cant factor

9. (x + 9)(x + 6)

10. (a + 5)(a + 7)

11. (b – 6)(b – 7)

12. (x + 2)(x + 13)

13. (a – 7)(a + 3)

14. cant factor

15. (x – 8)(x + 7)

16. (g + 7)(g + 2)

17. (x – 10)(x + 11)

18. (x + 8)(x + 3)

19. cant factor

20. (x + 7)(x – 3)

21. (y – 2)(y – 6)

22. cant factor

23. (x – 5)(x + 4)

24. (x – 9)(x + 8)

25. (x + 5)(x - 1)

26. (x + 5)(x + 10)

27. (x + 8)(x – 4)

28. (x + 6)(x + 1)

29. (x + 11)(x + 1)

30. (x + 10)(x + 2)

31. (x + 7)(x – 5)

32. (x – 6)(x – 12)

33. (x – 8)(x – 7)

34. (x – 8)(x + 2)

35. (x – 3)(x – 5)

36. (x – 8)(x + 9)

37. (x –13)(x – 3)

38. (x + 11)(x + 11)

39. (x + 12)(x + 1)

40. (x – 2y)(x – y)

41. (x – 12y)(x - 2y)

42. (x + 3y)(x + 2y)

43. (x – 7y)(x + 9y)

44. (x – 3y)(x + 11y)

45. (x – 4)(x – 4)

46. (x – 10)(x – 2)

47. (x – 11) (x – 1)

48. (c + 5)(c – 4)

49. (x + 6)(x + 6)

50. (x – 3)(x + 2)

51. (x + 5)(x + 7)

52. (x – 3)(x – 6)

53. (y – 6)(y – 7)

54. (x + 10)(x – 4)

55. (x + 12)(x – 11)

56. cant factor

57. (a – 12b)(a + 2b)

58. (m – 2n)(m – n)

59. (x + 11y)(x + 4y)

60. (t + 2)(t + 21)

61. (y – 6)(y – 6)

62. (b + 5)(b – 9)

63. (x – 3)(x + 6)

64. (c – 7)(c – 3)

Complex Trinomial Factoring

This method of factoring can only be used if you have trinomial expression where the leading coefficient

is greater than 1. i.e. 62 2 xx .

When using this method you need two find two numbers that multiply to give you the product of the

leading coefficient and the ending coefficient and add to give you the middle coefficient. Those two

numbers replace the middle term in the original expression. You then factor out a number from the first

two terms and another number from the last two terms. You should end up with the exact same two

brackets, and you factor out the common bracket to give you the final answer. It may be useful to set up

a diagram to organize your thoughts.

Example 1: Factor 62 2 xx

Solution: -12

)32)(2(

)2(3)2(2

6342

62

2

2

xx

xxx

xxx

xx


Solution: +36

)32)(23(

)23(3)23(2

6946

6136

2

2

xx

xxx

xxx

xx

+4 -3 = +1

+4 +9 = +13

Example 3: Factor 12194 2 aa

Solution: +48

)4)(34(

)34(4)34(

121634

12194

2

2

aa

aaa

aaa

aa


Solution: cannot factor by complex trinomial method, there are no two numbers that

multiply to give you -4 and add to give you +6.

Example 5: Factor 5194 2 zz

Solution: -20

)14)(5(

)5(1)5(4

51204

5194

2

2

zz

zzz

zzz

zz

-3 -16 = -19

-20 1 = -19

Factoring: Complex trinomials

1. 7203 2 xx

2. 352 2 xx

3. 5214 2 xx

4. 12285 2 xx

5. 372 2 xx

6. 64325 2 xx

7. 734 2 xx

8. 143 2 xx

9. 45192 2 xx

10. 56133 2 xx

11. 21254 2 xx

12. 54395 2 xx

13. 30416 2 xx

14. 2732 2 xx

15. 583 2 xx

16. 274 2 xx

17. 45114 2 xx

18. 30315 2 xx

19. 3532 2 xx

20. 42113 2 xx

21. 28395 2 xx

22. 9134 2 xx

23. 6173 2 xx

24. 63525 2 xx

25. 9556 2 xx

26. 7272 2 xx

27. 49283 2 xx

28. 8395 2 xx

29. 2894 2 xx

30. 30233 2 xx

31. 15132 2 xx

32. 12115 2 xx

33. 34 2 xx

34. 9283 2 xx

ANSWERS:

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

2. (2x + 1)(x – 3)

3. (4x + 1)(x + 5)

4. (5x + 2)(x – 6)

5. (x – 3)(2x – 1)

6. (x + 8)(5x – 8)

7. (4x – 7)(x + 1)

8. (3x + 1)(x + 1)

9. (2x + 9)(x + 5)

10. (3x + 8)(x – 7)

11. (4x + 3)(x – 7)

12. (5x + 6)(x – 9)

13. (6x – 5)(x – 6)

14. (x + 3)(2x – 9)

15. (x + 1)(3x + 5)

16. (4x – 1)(x + 2)

17. (4x + 9)(x – 5)

18. (x + 5)(5x + 6)

19. (2x + 7)(x – 5)

20. (3x + 7)(x – 6)

21. (5x – 4)(x – 2)

22. (4x – 9)(x – 1)

23. (3x – 1)(x + 6)

24. (5x + 7)(5x + 9)

25. (6x – 1)(x – 9)

26. (2x + 9)(x – 8)

27. (x + 7)(3x + 7)

28. (5x – 1)(x + 8)

29. (4x + 7)(x – 4)

30. (3x – 5)(x – 6)

31. (2x + 3)(x + 5)

32. (x – 3)(5x + 4)

33. (4x + 3)(x – 1)

34. (3x + 1)(x + 9)

Factoring: All Types

Most of the time the question will not tell you what type of factoring you should use, and sometimes

you need to factor the expression using more than one method. A good rule of thumb is to try to factor

using the GCF method first, and then determine if any other type of factoring can be used to further

break down the expression.

Example 1: Factor completely 36244 2 xx

Solution: First, look for a GCF, which there is of 4.

)96(4 2 xx

Second, see if the bracket can be factored using any other method, which

it can, by the simple trinomial method.

)3)(3(4 xx

Third, see if the new brackets can be factored, which they cannot. So, you

are done! The final answer is: )3)(3(4 xx

Example 2: Factor completely 25102 yy

Solution: First, look for a GCF, which there is none.

Second, see if it can be factored using any other method, which it can, by

the simple trinomial method.

)5)(5( yy

Third, see if the brackets can be factored, which they cannot. So, you are

done! The final answer is: )5)(5( yy

Example 3: Factor completely 149 2 x



it can, by the difference of squares method.

)17)(17( xx

Third, see if the brackets can be factored, which they cannot. So, you are

done! The final answer is: )17)(17( xx

Example 4: Factor completely xxx 21714 23

Solution: First, look for a GCF, which there is of 7x.

)32(7 2 xxx


they cannot (it looks like a complex trinomial, but that method doesn’t

work). So, you are done! The final answer is: )32(7 2 xxx




it can, by the complex trinomial method.

)4)(38( xx


are done! The final answer is: )4)(38( xx

Example 6: Factor completely 163212 2 zz

Solution: First, look for a GCF, which there is of 4.

)483(4 2 zz


it can, by the complex trinomial method. (make sure to keep the 4 you

factored out in step 1)

)23)(2(4 zz


are done! The final answer is: )23)(2(4 zz


Solution: First, look for a GCF, which there is of -2.

)183(2 2 xx


it can, by the simple trinomial method. (make sure to keep the -2 you

factored out in step 1)

)6)(3(2 xx


are done! The final answer is: )6)(3(2 xx

Factoring: all types

Completely factor the following expressions using the appropriate method.

1) 2a2b – 4ab2 2) 4 – x2

3) a2 + 8a + 16 4) 2a2 + 6a – ab – 3b

5) 9x2 + 100y2 6) 2y3 – 128

7) z2 – 7z + 12 8) y(3x – 2) + 4(2 – 3x)

9) 16c2 – 36 10) y2 + 5y – 24

11) m2 + 14m + 49 12) 8x4 – 4x3 + 12x2

13) 128t4 – 2t2 14) 27h3 + 8

15) 8x2 + 6x – 9 16) 4x3 + 12x2 – 8x

17) 14x3 – 7x2 - 21x 18) x2 + 2x – 35

19) 3x2 – 5x – 2 20) x2 – 18x + 81

21) 4x2 – 25y2 22) x2 – 6x – 16

23) 4y2 + 7y – 2 24) 5a3 – 25a2 + 15a

25) 9x2 – 16 26) x2 – 64

27) a2 + 12a + 27 28) 6x2 + 12x + 6

29) 2x3 - 20x2 – 48x 30) 12x5 – 6x3 + 3x2

31) 6y2 – 54 32) 4y2 – 17y - 15

33) 5x2 – 5 34) y5 + 3y3 + 4y2 + 12

35) x2 – 7x - 18 36) x2 – 8x + 16

37) 3x2 – 4x - 32 38) 4x2 + 2x - 20

39) 3x2 – 20x -7 40) 6x2 + x - 12

Answers:

1. 2ab(a – 2b)

2. (2 – x)(2 + x)

3. (a + 4)(a + 4)

4. (2a – b)(a + 3)

5. NF

6. 2(y3 – 64)

7. (z – 4)(z – 3)

8. (y – 4)(3x – 2)

9. (4c – 6)(4c + 6)

10. (y + 8)(y – 3)

11. (m+ 7)(m + 7)

12. 4x2(2x2 – x + 3)

13. 2t2(8t – 1)(8t + 1)

14. NF

15. (2x + 3)(4x – 3)

16. 4x(x2 + 3x – 2)

17. 7x(2x – 3)(x + 1)

18. (x – 5)(x + 7)

19. (3x + 1)(x – 2)

20. (x – 9)(x – 9)

21. (2x – 5y)(2x + 5y)

22. (x – 8)(x + 2)

23. (4y – 1)(y +2)

24. 5a(a2 – 5a + 3)

25. (3x – 4)(3x + 4)

26. (x – 8)(x + 8)

27. (a + 3)(a + 9)

28. 6(x + 1)(x + 1)

29. 2x(x – 12)(x + 2)

30. 3x2(4x3 – 2x + 1)

Documents

Factoring Review - manhassetschools.org€¦ · Factoring Review In Grade 10, you learned many ways of factoring: 1. GCF factoring 2. Difference of Squares factoring 3. Simple trinomial