1. Example 1 Factor ax2 + bx + c
Example 2 Factor When a, b, and c Have a Common Factor
Example 3 Determine Whether a Polynomial Is Prime
Example 4 Solve Equations by Factoring
Lesson 4 Contents

2. Example 1
Factor ax2 + bx + c

3. Step 1: Draw a box with 9 squares
Factor
In order to factor a trinomial with an “a” value we will use a technique called the factor box.
Step 1: Draw a box with 9 squares
Step 2: Place the first term in the top left box.
5x2 10
Step 3: Place the third term in the top middle box.
Example 4-1a

4. Step 5: Place the middle term outside the box at the right.
Factor
In a factor box, the left box times the middle box will equal the right box.
Step 4: 5x2 x 10 = 50x2
Step 5: Place the middle term outside the box at the right.
Step 6: Find the factors of 50x2 that add to 27x.
5x2 10
50x2
27x
Example 4-1a

5. Factor
Step 7: Place the 2 terms that multiply to be 50x2 and add to be 27x in the far right boxes.
5x2 10
50x2
27x 2x
25x
Example 4-1a

6. What can I multiply to get 2x?
Factor
Before looking at the 4 remaining boxes, remember the rules we have learned thus far: ) you multiply from left to right and the product is in the far right box ) you multiply from bottom to top and the result is in the top box.
Ask yourself:
What can I multiply to get 2x?
What can I multiply to get 25x?
As you are figuring that out, you also have to figure out how to multiply those factors going up to obtain 5x2 and 10.
Example 4-1a

7. Factor
Step 8: Place the 2 terms that multiply to be 2x in the top 2 remaining boxes.
Step 9: Place the 2 terms that multiply to be 25x in the bottom 2 remaining boxes.
5x2 10
50x2
27x 2x
25x x 2 5x 5
Example 4-1a

8. Vertical Check: Does x times 5x equal 5x2?
Factor
Vertical Check: Does x times 5x equal 5x2?
Does 5 x 2 = 10?
5x2
27x x 5x 10 2 5
Example 4-1a

9. Factor
Horizontal Check: Does x times 2 equal 2x?
Does 5x times 5 = 25x?
27x x 2 2x 5x 5
25x
Once everything checks out, you are ready to identify your final answer.
Example 4-1a

10. Factor
Your factors are in blue. Circle them diagonally. If there is no sign it is understood to be addition.
27x x 2 2x 5x 5
25x
5x2 10
50x2
Answer: (x + 5)(5x + 2)
Example 4-1a

11. Factor
26x
3x2 35
105x2 x 5 3x 7 5x
21x
Answer:
Example 4-1a

12. Factor
Look for the factors of 72x2 (24x2 times 3) that add to be -22x.
–73 –38 –27 –22
–1, –72 –2, –36 –4, –24 –4, –18
Sum of Factors
Factors of 72
The correct factors are –4, –18.
Example 4-1b

13. Factor 24x2 - 22x + 3
Now place the 2 terms that multiply to be 72x2 and add to be -22x in the far right boxes.
24x2 3
72x2
-22x
-4x
-18x
Example 4-1b

14. Factor 24x2 - 22x + 3
Next: Place the 2 terms that multiply to be -4x in the top 2 remaining boxes.
Then: Place the 2 terms that multiply to be -18x in the bottom 2 remaining boxes.
24x2 3
72x2
-22x
-4x
-18x 4x -1 6x -3
Example 4-1b

15. Factor 24x2 - 22x + 3
Your factors are in blue. Circle them diagonally. If there is no sign it is understood to be addition.
-22x 4x -1
-4x 6x -3
-18x
24x2 3
72x2
Answer: (4x - 3)(6x - 1)
Example 4-1b

16. a. Factor
-17x
3x2 10
30x2 x -2 3x -5
-2x
-15x
Answer:
Example 4-1b

17. b. Factor
-23x
10x2 12
120x2 2x -4 5x -3
-8x
-15x
Answer:
Example 4-1b

18. Example 2
Factor When a, b, and c
Have a Common Factor

19. First factor out the common term.
4(x2 + 6x + 8)
Now, factor the trinomial x2 + 6x + 8
Answer:
Example 4-2a

20. First factor out the common term.
2(x2 + 7x + 10)
Now, factor the trinomial x2 + 7x + 10
Answer:
Example 4-2b

21. Determine Whether a Polynomial Is Prime
Example 3
Determine Whether a Polynomial Is Prime

22. Factor
Look for the factors of -15x2 (3x2 times -5) that add to be 7x.
14 –14 2 –2
–1, 15 1, –15 –3, 5 3, –5
Sum of Factors
Factors of –15
Example 4-3a

23. Answer: is a prime polynomial.
There are no factors whose sum is 7. Therefore, cannot be factored using integers.
Answer: is a prime polynomial.
Example 4-3a

24. Factor
First make a table of the terms that multiply to be 9 but add to be -5.
–1, -9 1, 9 –3, -3 3, 3
Sum of Factors
Factors of 9
There are NO such factors so it canNOT be factored.
Answer: prime
Example 4-3b

25. Solve Equations by Factoring
Example 4
Solve Equations by Factoring

26. Hint: Key word SOLVE (set equation = 0)
Original equation
Hint: Key word SOLVE (set equation = 0)
Rewrite so one side equals 0.
Hint: Use the factor box to factor!
-14b
15b2 -8
-120b2 3b 2 5b -4 6b
-20b
Example 4-4a

27. Answer: The solution set is
Factor the left side. or
Zero Product Property
Solve each equation.
Answer: The solution set is
Example 4-4a

28. Hint: Use the factor box to factor!
Solve: 12x2 - 19x + 5 = 0
Original equation
12x2 - 19x + 5 = 0
Hint: Use the factor box to factor!
-19x
12x2 5
60x2 3x -5 4x -1
-15x
-4x
Example 4-4b

29. Factor the left side. (3x - 1)(4x - 5) = 0 or Zero Product Property
Solve each equation.
3x = 1
4x = 5
x = 1/3 x = 5/4
Answer:
Example 4-4b

30. Homework: Page even
Use the factor box to assist you with factoring.
Example 4-4b