Factoring Revie · 2017-07-07 · ... (x and y) is defined as the largest facto r that can be...
Transcript of Factoring Revie · 2017-07-07 · ... (x and y) is defined as the largest facto r that can be...
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.
Example 1: Factor 492 x
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
Example 4: Factor 493 x
Solution: cannot use DOS factoring because 3x is not a perfect square
Example 5: Factor 366 x
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
Example 2: Factor 6136 2 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
Example 4: Factor 262 2 xx
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
Solution: First, look for a GCF, which there is none.
Second, see if the bracket can be factored using any other method, which
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
Second, see if the bracket can be factored using any other method, which
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
Example 5: Factor completely 12298 2 xx
Solution: First, look for a GCF, which there is none.
Second, see if the bracket can be factored using any other method, which
it can, by the complex trinomial method.
)4)(38( xx
Third, see if the new brackets can be factored, which they cannot. So, you
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
Second, see if the bracket can be factored using any other method, which
it can, by the complex trinomial method. (make sure to keep the 4 you
factored out in step 1)
)23)(2(4 zz
Third, see if the new brackets can be factored, which they cannot. So, you
are done! The final answer is: )23)(2(4 zz
Example 7: Factor completely 3662 2 xx
Solution: First, look for a GCF, which there is of -2.
)183(2 2 xx
Second, see if the bracket can be factored using any other method, which
it can, by the simple trinomial method. (make sure to keep the -2 you
factored out in step 1)
)6)(3(2 xx
Third, see if the new brackets can be factored, which they cannot. So, you
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)