The Factoring Puzzle

1
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
2
4x38x22x

• GCF?
• Yes
• 2x
• Factor it out
• 2x(2x24x1)
• Factor further?
• All done.

3
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
4
x2-64

• GCF?
• No.
• Difference of Squares?
• Yes.
• Find the square roots of 1st last terms
• (x8)(x-8)
• More?
• No all done.

5
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
6
x218x81

• GCF?
• No.
• Difference of squares?
• No.
• Factor?
• What are the factors of the 1st last terms?

7
x218x81 (contd)

• X 1,81 9,9 4, 27
• (x _ _)(x _ _)
• 81 means?
• Both the same sign
• 18x means?
• Both must be
• (x_)(x_)

8
x218x81 (contd)

• Then what?
• (x_)(x_)
• (x9)(x9)

9
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
10
x2-4x4

• GCF?
• No.
• Difference of squares?
• No.
• Factor.
• Factors of 1st last terms
• x 2,2 1,4

11
x2-4x4 (contd)

• (x_ _)(x_ _)
• 4 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
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
13
3x26x3

• GCF?
• 3
• 3(x22x1)
• Factor Further?
• Yes.
• Factor x22x1
• (x _ _)(x _ _)

14
3x26x3 (contd)

• 1 means?
• Both signs are the same
• 2x means?
• Both factors must be sums ()
• Find the factors
• (x 1)(x 1)
• Final Answer 3(x1)(x1)

15
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
16
2x213x-7

• GCF?
• No.
• Difference of squares?
• No.
• Factor
• Factors of 2x2 are 2x,x

17
2x213x-7 (contd)

• (2x _ _)(x _ _)
• Factors of 7 1,7
• -7 means?
• One , one -
• 13x means?
• Larger product must be
• Try it...

18
2x213x-7 (contd)

• (2x1)(x-7)
• Outer Product -14x
• Inner Product 1x
• Sum -13x
• (2x-1)(x7)
• Outer Product 14x
• Inner Product -1x
• Sum 13x

19
2x213x-7 (contd)

• (2x1)(x-7)
• Outer Product -14x
• Inner Product 1x
• Sum -13x
• (2x-1)(x7) This is the one!
• Outer Product 14x
• Inner Product -1x
• Sum 13x

20
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
21
4x3-36x

• GCF?
• 4x
• 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 (contd)

• Roots x, 3
• 4x(x _ 3)(x _ 3)
• 4x(x3)(x-3)
• All done!

23
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
24
3x2-7x1

• GCF?
• No.
• Difference of squares?
• No.
• Factor?
• 3x,x 1,1
• Try the various combinations...

25
3x2-7x1 (contd)

• (3x1)(x-1)
• Products -3x, 1x
• Sum -2x
• (3x-1)(x1)
• Products 3x, -1x
• Sum 2x
• No matches, therefore 3x2-7x1 is prime.

26
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10
27
3x2-11x10

• GCF?
• No.
• Difference of squares?
• No.
• Factors
• 3x, x 1,10 2,5
• Try them...

28
3x2-11x10 (contd)

• (3x-1)(x-10)
• Products -30x, -x
• Sum -31x
• (3x-10)(x-1)
• Products -3x, -10x
• Sum -13x

29
3x2-11x10 (contd)

• (3x-2)(x-5)
• Products -15x, -2x
• Sum -17x
• (3x-5)(x-2)
• Products -6x, -5x
• Sum -11x

30
3x2-11x10 (contd)

• (3x-2)(x-5)
• Products -15x, -2x
• Sum -17x
• (3x-5)(x-2) This is the one!
• Products -6x, -5x
• Sum -11x
• It works!

31
The Factoring Puzzle
4x38x22x x2-64 x218x81
x2-4x4 3x26x3 2x213x-7
4x3-36x 3x2-7x1 3x2-11x10