Factoring Quadratics: Simple Trinomial

1. Factoring Quadratics:Simple Trinomial

2. Recall , expanding gives the following result:

3. The answer is called a Simple Trinomial because it fits the format given by ax2+bx+c and a =1.

4. Since factoring is the opposite of expanding, this means that the Simple Trinomialfrom the previous slide can be factored as follows: Note: The numbers -7 and 3 that occur in the factors play a role in creating the values in the Simple Trinomial , namely -7 x 3 = -21 and -7 + 3 = -4.

5. When factoring a Simple Trinomial, the quadratic expression must be recognized as having three terms in the form ax2+bx+c with a = 1. Example 1: Identify the Simple Trinomials.

6. Solution: Parts b) and d) are both Simple Trinomials, but part a) is not because a = 5 and part c) is not because it is not in the correct form as there are 3 different variables.

7. To factor a Simple Trinomial, the following general rule is used: where pxq = cand p+q = b.

8. For example, to factor k2+3k−40, two numbers must be found that multiply to -40 and add to +3. This is written as follows:

9. The factors of 40 are listed and tested to see which two factors can give the desired results of -40 and +3. This is usually done mentally or written down. When a factor list is written down, the unusable factors are lightly crossed out to avoid repeat guesses. Negative signs are inserted, if needed.

10. Since -5 and 8 give the desired results of -40 and +3, these are the numbers that will appear in the two factors. -5 8 -5 8 This means the factoring can now be completed as follows:

11. Example 2: Factor fully, when possible.

12. Solution: (1) Set up the product and sum needed here.

13. Solution cont’d: (2) Now list the factors of 10 until the usable factors are found. Remember to insert any needed negative signs.

14. Solution cont’d: (3) Fill the slots with the usable factors to check that both conditions are met.

15. -2 -5 -2 -5 Solution cont’d: (4) Now factor the given Simple Trinomial!

16. Try part b).

17. Solution cont’d: (1) Set up the product and sum needed here.

18. Solution cont’d: (2) Now list the factors of 24 until the usable factors are found. Remember to insert any needed negative signs.

19. Solution cont’d: (3) Fill the slots with the usable factors to check that both conditions are met.

20. -3 8 -3 8 Solution cont’d: (4) Now factor the given Simple Trinomial!

21. Solution cont’d:

22. It’s time to try part c).

23. Solution cont’d: • Set up the product and sum needed here. • List the needed factors until the usable factors are found. Remember to insert any needed negative signs. • Fill the slots with the usable factors to check that both conditions are met. • Factor this Simple Trinomial!

24. Solution cont’d: Here’s a possible solution. 6 8 6 8 The work above gives:

25. Finish by trying the following trickier questions: Example 3: Factor fully, if possible. Hint: Rewrite each question in the correct order and check for any common factors.

26. -1 -9 -1 -9 Solution:

27. -3 4 -3 4 Solution:

28. Solution: No usable factors here, so this Simple Trinomial CANNOT be factored!!