Factoring Trinomials III
Factoring Trinomials III
Too factor trinomials of the form ax2 + bx + c, the shortest,
but not a reliable way, is by guess...
Factoring Trinomials III
Reversed FOIL Method
Too factor trinomials of the form ax2 + bx + c, the shortest,
but not a reli...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions.
Reversed FOIL Method...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
For this method, we need to find four numbers that fit certain
descriptions. The following are ex...
Factoring Trinomials III
Too factor trinomials of the form ax2 + bx + c, the shortest,
but not a reliable way, is by guess...
Factoring Trinomials III
(Reversed FOIL Method)
Factoring Trinomials III
(Reversed FOIL Method)
Let’s see how the above examples are related to factoring.
Example B. Factor 3x2 + 5x + 2.
Factoring Trinomials III
(Reversed FOIL Method)
Let’s see how the above examples are relat...
Example B. Factor 3x2 + 5x + 2.
The only way to get 3x2 is (3x ± #)(1x ± #).
Factoring Trinomials III
(Reversed FOIL Metho...
Example B. Factor 3x2 + 5x + 2.
The only way to get 3x2 is (3x ± #)(1x ± #).
The #’s must be 1 and 2
Factoring Trinomials ...
Example B. Factor 3x2 + 5x + 2.
The only way to get 3x2 is (3x ± #)(1x ± #).
The #’s must be 1 and 2 to get the constant t...
Example B. Factor 3x2 + 5x + 2.
The only way to get 3x2 is (3x ± #)(1x ± #).
The #’s must be 1 and 2 to get the constant t...
Example B. Factor 3x2 + 5x + 2.
The only way to get 3x2 is (3x ± #)(1x ± #).
The #’s must be 1 and 2 to get the constant t...
3(± # ) +1(± #) = 5 where the #’s are 1 and 2.
Factoring Trinomials III
(Reversed FOIL Method)
Let’s see how the above exa...
3(± # ) +1(± #) = 5 where the #’s are 1 and 2.
Since 3(1) +1(2) = 5,
Factoring Trinomials III
(Reversed FOIL Method)
Let’s...
3(± # ) +1(± #) = 5 where the #’s are 1 and 2.
Since 3(1) +1(2) = 5,
Factoring Trinomials III
(Reversed FOIL Method)
Let’s...
3(± # ) +1(± #) = 5 where the #’s are 1 and 2.
Since 3(1) +1(2) = 5,
Factoring Trinomials III
(Reversed FOIL Method)
Let’s...
3(± # ) +1(± #) = 5 where the #’s are 1 and 2.
Since 3(1) +1(2) = 5, we see that
3x2 + 5x + 2 = (3x + 2)(1x + 1).
Factorin...
3(± # ) +1(± #) = 5 where the #’s are 1 and 2.
Since 3(1) +1(2) = 5, we see that
3x2 + 5x + 2 = (3x + 2)(1x + 1).
5x
Facto...
Factoring Trinomials III
Example C. Factor 3x2 – 7x + 2.
Factoring Trinomials III
Example C. Factor 3x2 – 7x + 2.
We start with (3x ± #)(1x ± #).
Factoring Trinomials III
Example C. Factor 3x2 – 7x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1 and 2 as #'s
3(± # ) + 1(± # ) = –7.
Factoring Trinomials III
Example C. Factor 3x2 – 7x + 2.
We start with (3x ± #)(1x ± #).
We need t...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
Factoring Trinomials III
Example C. Factor 3x2 – 7x + 2.
We start with (3...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Factoring Trinomials III
Example C. Fac...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
Factori...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
We star...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
We star...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
We star...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
We star...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
We star...
3(± # ) + 1(± # ) = –7.
It's 3(–2) + 1(–1) = –7.
So 3x2 – 7x + 2 = (3x –1)(1x – 2)
Example D. Factor 3x2 + 5x – 2.
We star...
Example E. Factor 3x2 + 8x + 2.
Factoring Trinomials III
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
Factoring Trinomials III
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Fac...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Example E. Factor 3x2 + 8x + 2.
We start with (3x ± #)(1x ± #).
We need to fill in 1&2 so that
3(± # ) + 1(± # ) = +8.
Thi...
Factoring Trinomials III
It's not necessary to always start with ax2. If c is a prime
number, we start with c.
Example G. Factor 12x2 – 5x – 3.
Factoring Trinomials III
It's not necessary to always start with ax2. If c is a prime
num...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1),
Factoring Trinomials III
It's not necessary to always start with ax...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12
Factoring Trinomi...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
Factorin...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
Factorin...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
Factorin...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
Factorin...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
So 12x2 ...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
So 12x2 ...
Example G. Factor 12x2 – 5x – 3.
Since 3 must be 3(1), we need to find two numbers such
that (#)(#) = 12 and that
So 12x2 ...
Example H. Factor 3x2 – 7x – 2 .
Factoring Trinomials III
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #).
Factoring Trinomials III
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
Factoring Trinomials III
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Example H. Factor 3x2 – 7x – 2 .
We start with (3x ± #)(1x ± #). We find that:
3(–2) + 1(–1) = –7.
But this won't work sin...
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order.
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Factoring Trinomials III
Finally, before starting the reverse-FOIL procedure
1. make sure the terms are arranged in order....
Ex. A. Factor the following trinomials. If it’s prime, state so.
1. 3x2 – x – 2 2. 3x2 + x – 2 3. 3x2 – 2x – 1
4. 3x2 + 2x...
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5 5factoring trinomial iii

  1. 1. Factoring Trinomials III
  2. 2. Factoring Trinomials III Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  3. 3. Factoring Trinomials III Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  4. 4. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  5. 5. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  6. 6. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  7. 7. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  8. 8. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Yes, 1* (2) + 3 * (1) = 5 Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  9. 9. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Yes, 1* (2) + 3 * (1) = 5 b. 1* (± ) + 3* (± ) = –5. Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  10. 10. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Yes, 1* (2) + 3 * (1) = 5 b. 1* (± ) + 3* (± ) = –5. Yes, 1* (1) + 3* (–2) = –5 Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  11. 11. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Yes, 1* (2) + 3 * (1) = 5 b. 1* (± ) + 3* (± ) = –5. Yes, 1* (1) + 3* (–2) = –5 or 1* (–2) + 3* (–1) = –5 Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  12. 12. Factoring Trinomials III For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Yes, 1* (2) + 3 * (1) = 5 b. 1* (± ) + 3* (± ) = –5. Yes, 1* (1) + 3* (–2) = –5 or 1* (–2) + 3* (–1) = –5 c. 1* (± ) + 3* (± ) = 8. Reversed FOIL Method Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed.
  13. 13. Factoring Trinomials III Too factor trinomials of the form ax2 + bx + c, the shortest, but not a reliable way, is by guessing, .i.e. we reverse the FOIL procedure and try to guess what numbers are needed. For this method, we need to find four numbers that fit certain descriptions. The following are examples of the task to be accomplished. Example A. Let {1, 3} and {1, 2} be two pairs of numbers. Is it possible to split the {1, 2 }, put them in the boxes that makes the equality true? a. 1* (± ) + 3*(± ) = 5. Yes, 1* (2) + 3 * (1) = 5 b. 1* (± ) + 3* (± ) = –5. Yes, 1* (1) + 3* (–2) = –5 or 1* (–2) + 3* (–1) = –5 c. 1* (± ) + 3* (± ) = 8. No, since the most we can obtain is 1* (1) + 3* (2) = 7. Reversed FOIL Method
  14. 14. Factoring Trinomials III (Reversed FOIL Method)
  15. 15. Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  16. 16. Example B. Factor 3x2 + 5x + 2. Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  17. 17. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  18. 18. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  19. 19. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  20. 20. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  21. 21. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring.
  22. 22. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that
  23. 23. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that
  24. 24. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that
  25. 25. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that
  26. 26. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, we see that 3x2 + 5x + 2 = (3x + 2)(1x + 1). Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that
  27. 27. 3(± # ) +1(± #) = 5 where the #’s are 1 and 2. Since 3(1) +1(2) = 5, we see that 3x2 + 5x + 2 = (3x + 2)(1x + 1). 5x Factoring Trinomials III (Reversed FOIL Method) Let’s see how the above examples are related to factoring. Example B. Factor 3x2 + 5x + 2. The only way to get 3x2 is (3x ± #)(1x ± #). The #’s must be 1 and 2 to get the constant term +2. We need to place 1 and 2 as the #'s so the product will yield the correct middle term +5x. That is, (3x ± #)(1x ± #) must yields +5x, or that
  28. 28. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2.
  29. 29. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #).
  30. 30. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s
  31. 31. 3(± # ) + 1(± # ) = –7. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  32. 32. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  33. 33. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  34. 34. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  35. 35. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. We start with (3x ± #)(1x ± #). Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  36. 36. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  37. 37. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that 3(± # ) + 1(± # ) = +5. Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  38. 38. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that 3(± # ) + 1(± # ) = +5. Since c is negative, they must have opposite signs . Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  39. 39. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that 3(± # ) + 1(± # ) = +5. It is 3(+2) + 1(–1) = +5. Since c is negative, they must have opposite signs . Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  40. 40. 3(± # ) + 1(± # ) = –7. It's 3(–2) + 1(–1) = –7. So 3x2 – 7x + 2 = (3x –1)(1x – 2) Example D. Factor 3x2 + 5x – 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that 3(± # ) + 1(± # ) = +5. It is 3(+2) + 1(–1) = +5. So 3x2 + 5x + 2 = (3x –1)(1x + 2) Since c is negative, they must have opposite signs . Factoring Trinomials III Example C. Factor 3x2 – 7x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1 and 2 as #'s so that
  41. 41. Example E. Factor 3x2 + 8x + 2. Factoring Trinomials III
  42. 42. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). Factoring Trinomials III
  43. 43. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. Factoring Trinomials III
  44. 44. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Factoring Trinomials III
  45. 45. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. Factoring Trinomials III
  46. 46. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check.
  47. 47. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4.
  48. 48. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #).
  49. 49. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4),
  50. 50. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. 3(± # ) + 1(± # ) = +11. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), we need to fill in 2&2 or 1&4 as #'s so that
  51. 51. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. 3(± # ) + 1(± # ) = +11. It can't be 2&2. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), we need to fill in 2&2 or 1&4 as #'s so that 3(± # ) + 1(± # ) = +11.
  52. 52. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. 3(± # ) + 1(± # ) = +11. It can't be 2&2. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), we need to fill in 2&2 or 1&4 as #'s so that Try 1&4, 3(± # ) + 1(± # ) = +11.
  53. 53. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. 3(± # ) + 1(± # ) = +11. It can't be 2&2. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), we need to fill in 2&2 or 1&4 as #'s so that Try 1&4, it is 3(+4) + 1(–1) = +11. 3(± # ) + 1(± # ) = +11.
  54. 54. Example E. Factor 3x2 + 8x + 2. We start with (3x ± #)(1x ± #). We need to fill in 1&2 so that 3(± # ) + 1(± # ) = +8. This is impossible. Hence the expression is prime. 3(± # ) + 1(± # ) = +11. It can't be 2&2. Factoring Trinomials III If both the numbers a and c in ax2 + bx + c have many factors then there are many possibilities to check. Example F. Factor 3x2 + 11x – 4. We start with (3x ± #)(1x ± #). Since 4 = 2(2) = 1(4), we need to fill in 2&2 or 1&4 as #'s so that Try 1&4, it is 3(+4) + 1(–1) = +11. So 3x2 + 11x – 4 = (3x – 1)(1x + 4).
  55. 55. Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c.
  56. 56. Example G. Factor 12x2 – 5x – 3. Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c.
  57. 57. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c.
  58. 58. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c.
  59. 59. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5.
  60. 60. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5. 12 = 1(12) = 2(6) = 3(4)
  61. 61. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5. 12 = 1(12) = 2(6) = 3(4) 1&12 and 2&6 can be quickly eliminated.
  62. 62. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5. 12 = 1(12) = 2(6) = 3(4) 1&12 and 2&6 can be quickly eliminated. We get (3)(–3) + (4)(+1) = – 5.
  63. 63. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that So 12x2 – 5x – 3 = (3x + 1)(4x – 3). Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5. 12 = 1(12) = 2(6) = 3(4) 1&12 and 2&6 can be quickly eliminated. We get (3)(–3) + (4)(+1) = – 5.
  64. 64. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that So 12x2 – 5x – 3 = (3x + 1)(4x – 3). Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5. 12 = 1(12) = 2(6) = 3(4) 1&12 and 2&6 can be quickly eliminated. We get (3)(–3) + (4)(+1) = – 5. Remark: In the above method, finding (#)(± #) + (#)( ± #) = b does not guarantee that the trinomial will factor.
  65. 65. Example G. Factor 12x2 – 5x – 3. Since 3 must be 3(1), we need to find two numbers such that (#)(#) = 12 and that So 12x2 – 5x – 3 = (3x + 1)(4x – 3). Factoring Trinomials III It's not necessary to always start with ax2. If c is a prime number, we start with c. (± #)(± 3) + (± #)(±1) = – 5. 12 = 1(12) = 2(6) = 3(4) 1&12 and 2&6 can be quickly eliminated. We get (3)(–3) + (4)(+1) = – 5. Remark: In the above method, finding (#)(± #) + (#)( ± #) = b does not guarantee that the trinomial will factor. We have to match the sign of c also.
  66. 66. Example H. Factor 3x2 – 7x – 2 . Factoring Trinomials III
  67. 67. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). Factoring Trinomials III
  68. 68. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. Factoring Trinomials III
  69. 69. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. Factoring Trinomials III
  70. 70. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. In fact this trinomial is prime. Factoring Trinomials III
  71. 71. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. In fact this trinomial is prime. Factoring Trinomials III There might be multiple matchings for (#)(± #) + (#)( ± #) = b make sure you chose the correct one, if any.
  72. 72. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. In fact this trinomial is prime. Factoring Trinomials III Example I: Factor 1x2 + 5x – 6 . There might be multiple matchings for (#)(± #) + (#)( ± #) = b make sure you chose the correct one, if any.
  73. 73. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. In fact this trinomial is prime. Factoring Trinomials III Example I: Factor 1x2 + 5x – 6 . We have: 1(+3) + 1(+2) = +5 There might be multiple matchings for (#)(± #) + (#)( ± #) = b make sure you chose the correct one, if any.
  74. 74. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. In fact this trinomial is prime. Factoring Trinomials III Example I: Factor 1x2 + 5x – 6 . We have: 1(+3) + 1(+2) = +5 1(+6) + 1(–1) = +5 There might be multiple matchings for (#)(± #) + (#)( ± #) = b make sure you chose the correct one, if any.
  75. 75. Example H. Factor 3x2 – 7x – 2 . We start with (3x ± #)(1x ± #). We find that: 3(–2) + 1(–1) = –7. But this won't work since (–2)(–1) = 2 = c. In fact this trinomial is prime. Factoring Trinomials III Example I: Factor 1x2 + 5x – 6 . We have: 1(+3) + 1(+2) = +5 The one that works is x2 + 5x – 6 = (x + 6)(x – 1). 1(+6) + 1(–1) = +5 There might be multiple matchings for (#)(± #) + (#)( ± #) = b make sure you chose the correct one, if any.
  76. 76. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order.
  77. 77. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first.
  78. 78. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first. 3. make sure that x2 is positive, if not, factor out the negative sign first.
  79. 79. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first. 3. make sure that x2 is positive, if not, factor out the negative sign first. Example J. Factor –x3 + 3x + 2x2
  80. 80. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first. 3. make sure that x2 is positive, if not, factor out the negative sign first. Example J. Factor –x3 + 3x + 2x2 –x3 + 3x + 2x2 Arrange the terms in order = –x3 + 2x2 + 3x
  81. 81. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first. 3. make sure that x2 is positive, if not, factor out the negative sign first. Example J. Factor –x3 + 3x + 2x2 –x3 + 3x + 2x2 Arrange the terms in order = –x3 + 2x2 + 3x Factor out the GCF = – x(x2 – 2x – 3)
  82. 82. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first. 3. make sure that x2 is positive, if not, factor out the negative sign first. Example J. Factor –x3 + 3x + 2x2 –x3 + 3x + 2x2 Arrange the terms in order = –x3 + 2x2 + 3x Factor out the GCF = – x(x2 – 2x – 3) = – x(x – 3)(x + 1)
  83. 83. Factoring Trinomials III Finally, before starting the reverse-FOIL procedure 1. make sure the terms are arranged in order. 2. if there is any common factor, pull out the GCF first. 3. make sure that x2 is positive, if not, factor out the negative sign first. Example J. Factor –x3 + 3x + 2x2 –x3 + 3x + 2x2 Arrange the terms in order = –x3 + 2x2 + 3x Factor out the GCF = – x(x2 – 2x – 3) = – x(x – 3)(x + 1) For factoring problems, try guessing first. If no answer is found, use the ac-method (or the formula) to determine if it is prime or do it by grouping, and always check your answers.
  84. 84. Ex. A. Factor the following trinomials. If it’s prime, state so. 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 Factoring Trinomials III 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. –15x2 – 25x2 – 10x 34. 12x2y2 –14x2y2 + 4xy2

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