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Factoring Trinomials II-the ac-method
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Factoring Trinomials II-the ac-me...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
Some four terms formulas may be factored by the
grouping method, i.e. pulling out twice.
Example A.
a. Factor 3x – 3y + ax...
ac-Method:
Factoring Trinomials II-the ac-method
ac-Method: We assume that there is no common factor for
the trinomial ax2 + bx + c.
Factoring Trinomials II-the ac-method
...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
ac-Method: We assume that there is no common factor for
the trinomial...
Example B. Factor 3x2 – 4x – 20 using the ac-method.
Because a = 3, c = –20, we’ve
ac = 3(–20) = –60. We need
two numbers ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20
u v
1 60
2 30
3 20
4 15
5 12
6 10
6*(–10) = – 60
6 + (–10) = –4
Factoring Trinomi...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20)
u v
1 60
2 30
3 20
4 15
5 12
6 10
6...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
In this section we give a formula that enables us to tell if a
trinomial is factorable or not.
Factoring Trinomials II-the...
In this section we give a formula that enables us to tell if a
trinomial is factorable or not.
This formula is an outcome ...
Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a
squared number, then the trinomial ax2 + bx + c is factorable....
Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a
squared number, then the trinomial ax2 + bx + c is factorable....
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
In this section we...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Example D. Use the b2 – 4ac to see if the trinomial is
factorable. If it is, factor it.
a. 3x2 – 7x – 2
b2 – 4ac = (–7)2 –...
Write 3x2 – 4x – 20
= 3x2 + 6x –10x – 20 put in two groups
= (3x2 + 6x ) + (–10x – 20) pull out common factor
= 3x(x + 2) ...
1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x – 1
4. 3x2 + 2x – 1 5. 2x2 – 3x + 1 6. 2x2 + 3x – 1
8. 2x2 – 3x – 27. 2x2 + 3x –...
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5 3 factoring trinomial ii

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5 3 factoring trinomial ii

  1. 1. Factoring Trinomials II-the ac-method
  2. 2. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Factoring Trinomials II-the ac-method
  3. 3. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. b. Factor x2 – x – 6 by grouping. Factoring Trinomials II-the ac-method
  4. 4. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) b. Factor x2 – x – 6 by grouping. Factoring Trinomials II-the ac-method
  5. 5. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) b. Factor x2 – x – 6 by grouping. Factoring Trinomials II-the ac-method
  6. 6. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) b. Factor x2 – x – 6 by grouping. Factoring Trinomials II-the ac-method
  7. 7. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) b. Factor x2 – x – 6 by grouping. We write x2 – x – 6 = x2 – 3x + 2x – 6 Factoring Trinomials II-the ac-method
  8. 8. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) b. Factor x2 – x – 6 by grouping. We write x2 – x – 6 = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factors = x(x – 3) + 2(x – 3) Factoring Trinomials II-the ac-method
  9. 9. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) b. Factor x2 – x – 6 by grouping. We write x2 – x – 6 = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factors = x(x – 3) + 2(x – 3) Take out the common (x – 3) = (x – 3)(x + 2) Factoring Trinomials II-the ac-method
  10. 10. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) b. Factor x2 – x – 6 by grouping. We write x2 – x – 6 = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factors = x(x – 3) + 2(x – 3) Take out the common (x – 3) = (x – 3)(x + 2) Factoring Trinomials II-the ac-method ?
  11. 11. Some four terms formulas may be factored by the grouping method, i.e. pulling out twice. Example A. a. Factor 3x – 3y + ax – ay by grouping. 3x – 3y + ax – ay Group them into two groups. = (3x – 3y) + (ax – ay) Factor out the GCF of each group. = 3(x – y) + a(x – y) Pull the factor (x – y) again. = (3 + a)(x – y) b. Factor x2 – x – 6 by grouping. We write x2 – x – 6 = x2 – 3x + 2x – 6 Put them into two groups = (x2 – 3x) + (2x – 6) Take out the common factors = x(x – 3) + 2(x – 3) Take out the common (x – 3) = (x – 3)(x + 2) We use the ac-method to write trinomials into four-term formulas for grouping. Factoring Trinomials II-the ac-method ?
  12. 12. ac-Method: Factoring Trinomials II-the ac-method
  13. 13. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. Factoring Trinomials II-the ac-method Example B. Factor 3x2 – 4x – 20 using the ac-method.
  14. 14. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, Factoring Trinomials II-the ac-method
  15. 15. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, Because a = 3, c = –20, we’ve ac = 3(–20) = –60. Factoring Trinomials II-the ac-method
  16. 16. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. Factoring Trinomials II-the ac-method
  17. 17. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Factoring Trinomials II-the ac-method
  18. 18. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method
  19. 19. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method –60 –4 1 60 u v 2, ,303, 15,12,45, , 20,
  20. 20. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method –60 –4 1 60 u v 2, ,303, , 20, 15,12,45,
  21. 21. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method –60 –4 1 60 u v 2, ,303, , 20, 15,12, 10 45, 6
  22. 22. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method u v 1 60 2 30 3 20 4 15 5 12 6 10 –60 –4 1 60 u v 2, ,303, , 20, 15,12, 10 45, 6
  23. 23. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method u v 1 60 2 30 3 20 4 15 5 12 6 10 6*(–10) = – 60 6 + (–10) = –4 –60 –4 1 60 u v 2, ,303, , 20, 15,12, 10 45, 6
  24. 24. Example B. Factor 3x2 – 4x – 20 using the ac-method. ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. 2. Write ax2 + bx + c as ax2 + ux + vx +c then use the grouping method to factor (ax2 + ux) + (vx + c). Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method u v 1 60 2 30 3 20 4 15 5 12 6 10 6*(–10) = – 60 6 + (–10) = –4 –60 –4 1 60 u v 2, ,303, , 20, 15,12, 10 45, 6
  25. 25. Example B. Factor 3x2 – 4x – 20 using the ac-method. Because a = 3, c = –20, we’ve ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –4. Here are two searching methods -by the X-table, or a regular table. Factoring Trinomials II-the ac-method u v 1 60 2 30 3 20 4 15 5 12 6 10 6*(–10) = – 60 6 + (–10) = –4 –60 –4 1 60 u v 2, ,303, , 20, 15,12, 10 45, 6 ac-Method: We assume that there is no common factor for the trinomial ax2 + bx + c. 1. Calculate ac, and find two numbers u and v such that uv is ac, and u + v = b. 2. Write ax2 + bx + c as ax2 + ux + vx +c then use the grouping method to factor (ax2 + ux) + (vx + c) If step 1 can’t be done, then the expression is prime.
  26. 26. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 u v 1 60 2 30 3 20 4 15 5 12 6 10 6*(–10) = – 60 6 + (–10) = –4 Factoring Trinomials II-the ac-method
  27. 27. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) u v 1 60 2 30 3 20 4 15 5 12 6 10 6*(–10) = – 60 6 + (–10) = –4 Factoring Trinomials II-the ac-method
  28. 28. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) u v 1 60 2 30 3 20 4 15 5 12 6 10 6*(–10) = – 60 6 + (–10) = –4 Factoring Trinomials II-the ac-method
  29. 29. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Factoring Trinomials II-the ac-method
  30. 30. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 by the ac-method, if possible. If it’s prime, use a table to justify your answer. Factoring Trinomials II-the ac-method
  31. 31. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 by the ac-method, if possible. If it’s prime, use a table to justify your answer. a = 3, c = –20, hence ac = 3(–20) = –60. Factoring Trinomials II-the ac-method
  32. 32. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 by the ac-method, if possible. If it’s prime, use a table to justify your answer. a = 3, c = –20, hence ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –6. Factoring Trinomials II-the ac-method
  33. 33. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 by the ac-method, if possible. If it’s prime, use a table to justify your answer. a = 3, c = –20, hence ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –6. u v 1 60 2 30 3 20 4 15 5 12 6 10 Factoring Trinomials II-the ac-method
  34. 34. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 by the ac-method, if possible. If it’s prime, use a table to justify your answer. a = 3, c = –20, hence ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –6. u v 1 60 2 30 3 20 4 15 5 12 6 10 Factoring Trinomials II-the ac-method After examining all possible pairs of u's and v’s, we see that no such u and v exists. no u and v such that uv = –60 and u + v = –6.
  35. 35. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 by the ac-method, if possible. If it’s prime, use a table to justify your answer. a = 3, c = –20, hence ac = 3(–20) = –60. We need two numbers u and v such that uv = –60 and u + v = –6. u v 1 60 2 30 3 20 4 15 5 12 6 10 Factoring Trinomials II-the ac-method After examining all possible pairs of u's and v’s, we see that no such u and v exists. no u and v such that uv = –60 and u + v = –6.Hence 3x2 – 6x – 20 must be prime.
  36. 36. In this section we give a formula that enables us to tell if a trinomial is factorable or not. Factoring Trinomials II-the ac-method
  37. 37. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method
  38. 38. Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial ax2 + bx + c is factorable. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method
  39. 39. Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial ax2 + bx + c is factorable. Otherwise, it is not factorable. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method
  40. 40. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  41. 41. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  42. 42. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  43. 43. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  44. 44. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  45. 45. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. b. 3x2 – 7x + 2 Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  46. 46. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. b. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  47. 47. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. b. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  48. 48. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. b. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  49. 49. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. b. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  50. 50. Example D. Use the b2 – 4ac to see if the trinomial is factorable. If it is, factor it. a. 3x2 – 7x – 2 b2 – 4ac = (–7)2 – 4(3)(–2) = 49 + 24 = 73 is not a square, hence it is prime. In this section we give a formula that enables us to tell if a trinomial is factorable or not. This formula is an outcome of the quadratic formula. b. 3x2 – 7x + 2 b2 – 4ac = (–7)2 – 4(3)(2) = 49 – 24 = 25 which is a squared number, hence it is factorable. In fact 3x2 – 7x + 2 = (3x – 1)(x – 2) Factoring Trinomials II-the ac-method Theorem: If b2 – 4ac = 0, 1, 4, 9, 16, 25, 36, .. i.e. is a squared number, then the trinomial is factorable. Otherwise, it is not factorable.
  51. 51. Write 3x2 – 4x – 20 = 3x2 + 6x –10x – 20 put in two groups = (3x2 + 6x ) + (–10x – 20) pull out common factor = 3x(x + 2) – 10 (x + 2) pull out common factor = (3x – 10)(x + 2) Example C. Factor 3x2 – 6x – 20 using the ac-method, if possible. a = 3, c = –20, hence ac = 3(-20) = –60. We need two numbers u and v such that uv = –60 and u + v = –6. After searching all possibilities we found that it's impossible. Hence 3x2 – 6x – 20 is prime. Factoring Trinomials II-the ac-method
  52. 52. 1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x – 1 4. 3x2 + 2x – 1 5. 2x2 – 3x + 1 6. 2x2 + 3x – 1 8. 2x2 – 3x – 27. 2x2 + 3x – 2 15. 6x2 + 5x – 6 10. 5x2 + 9x – 2 B. Factor. Factor out the GCF, the “–”, and arrange the terms in order first. 9. 5x2 – 3x – 2 12. 3x2 – 5x + 211. 3x2 + 5x + 2 14. 6x2 – 5x – 613. 3x2 – 5x + 2 16. 6x2 – x – 2 17. 6x2 – 13x + 2 18. 6x2 – 13x – 2 19. 6x2 + 7x + 2 20. 6x2 – 7x + 2 21. 6x2 – 13x + 6 22. 6x2 + 13x + 6 23. 6x2 – 5x – 4 24. 6x2 – 13x + 8 25. 6x2 – 13x – 8 25. 4x2 – 9 26. 4x2 – 49 27. 25x2 – 4 28. 4x2 + 9 29. 25x2 + 9 30. – 6x2 – 5xy + 6y2 31. – 3x2 + 2x3– 2x 32. –6x3 – x2 + 2x 33. –15x3 – 25x2 – 10x 34. 12x3y2 –14x2y2 + 4xy2 Exercise A. Use the ac–method, factor the trinomial or demonstrate that it’s not factorable. Factoring Trinomials II-the ac-method

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