1. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

2. 4x3+8x2+2x GCF? Factor it out… Factor further? Yes… 2x(2x2+4x+1)
All done.

3. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

4. x2-64 GCF? Difference of Squares?
No.
Difference of Squares?
Yes.
Find the square roots of 1st & last terms
(x+8)(x-8)
More?
No; all done.

5. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

6. x2+18x+81 GCF? Difference of squares? Factor?
No.
Difference of squares?
Factor?
What are the factors of the 1st & last terms?

7. x2+18x+81 (cont’d) X; 1,81; 9,9; 4, 27;… (x _ _)(x _ _) +81 means?
Both the same sign
+18x means?
Both must be +
(x+_)(x+_)

8. x2+18x+81 (cont’d)
Then what?
(x+_)(x+_)…
(x+9)(x+9)

9. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

10. x2-4x+4 GCF? Difference of squares? Factor.
No.
Difference of squares?
Factor.
Factors of 1st & last terms
x | 2,2; 1,4

11. x2-4x+4 (cont’d) (x_ _)(x_ _) +4 means? -4x means?
Both factors have the same sign
-4x means?
Both factors must be -
(x-_)(x-_)… Try it out…
(x-2)(x-2)

12. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

13. 3x2+6x+3 GCF? 3(x2+2x+1) Factor Further? Factor x2+2x+1 (x _ _)(x _ _)
Yes.
Factor x2+2x+1
(x _ _)(x _ _)

14. 3x2+6x+3 (cont’d) +1 means? +2x means? Find the factors…
Both signs are the same
+2x means?
Both factors must be sums (+)
Find the factors…
(x + 1)(x + 1)
Final Answer: 3(x+1)(x+1)

15. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

16. 2x2+13x-7 GCF? Difference of squares? Factor… Factors of 2x2 are 2x,x
No.
Difference of squares?
Factor…
Factors of 2x2 are 2x,x

17. 2x2+13x-7 (cont’d) (2x _ _)(x _ _) Factors of 7: 1,7 -7 means?
One +, one -
+13x means?
Larger product must be +
Try it...

18. 2x2+13x-7 (cont’d) (2x+1)(x-7) (2x-1)(x+7) Outer Product: -14x
Inner Product: +1x
Sum: -13x
(2x-1)(x+7)
Outer Product: +14x
Inner Product: -1x
Sum: +13x

19. 2x2+13x-7 (cont’d) (2x+1)(x-7) *(2x-1)(x+7)* This is the one!
Outer Product: -14x
Inner Product: +1x
Sum: -13x
*(2x-1)(x+7)* This is the one!
Outer Product: +14x
Inner Product: -1x
Sum: +13x

20. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

21. 4x3-36x GCF? 4x(x2-9) Factor Further?
Yes; is (x2-9) a difference of squares?
Yes
Find the square roots of the 1st & last terms

22. 4x3-36x (cont’d)
Roots: x, 3
4x(x _ 3)(x _ 3)
4x(x+3)(x-3)
All done!

23. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

24. 3x2-7x+1 GCF? Difference of squares? Factor?
No.
Difference of squares?
Factor?
3x,x | 1,1
Try the various combinations...

25. 3x2-7x+1 (cont’d) (3x+1)(x-1) (3x-1)(x+1)
Products: -3x, +1x
Sum: -2x
(3x-1)(x+1)
Products: +3x, -1x
Sum: +2x
No matches, therefore 3x2-7x+1 is prime.

26. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3

27. 3x2-11x+10 GCF? Difference of squares? Factors… Try them... No.

28. 3x2-11x+10 (cont’d) (3x-1)(x-10) (3x-10)(x-1) Products: -30x, -x
Sum: -31x
(3x-10)(x-1)
Products: -3x, -10x
Sum: -13x

29. 3x2-11x+10 (cont’d) (3x-2)(x-5) (3x-5)(x-2) Products: -15x, -2x
Sum: -17x
(3x-5)(x-2)
Products: -6x, -5x
Sum: -11x

30. 3x2-11x+10 (cont’d) (3x-2)(x-5) *(3x-5)(x-2)* This is the one!
Products: -15x, -2x
Sum: -17x
*(3x-5)(x-2)* This is the one!
Products: -6x, -5x
Sum: -11x
It works!

31. The Factoring Puzzle 4x3+8x2+2x x2-64 x2+18x+81 x2-4x+4 3x2+6x+3