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The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2...

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The Factoring Puzzle 4x 3 +8x 2 +2x x 2 -64 x 2 +18x+81 x 2 -4x+4 3x 2 +6x+3 2x 2 +13x-7 4x 3 -36x 3x 2 -7x+1 3x 2 -11x+10
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Page 1: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 2: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

4x3+8x2+2x

GCF?– Yes…

• 2x

Factor it out…– 2x(2x2+4x+1)

Factor further?– All done.

Page 3: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 4: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

x2-64

GCF?– No.

Difference of Squares?– Yes.

Find the square roots of 1st & last terms (x+8)(x-8) More?

– No; all done.

Page 5: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 6: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

x2+18x+81

GCF?– No.

Difference of squares?– No.

Factor? What are the factors of the 1st & last terms?

Page 7: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

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+_)

Page 8: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

x2+18x+81 (cont’d)

Then what? (x+_)(x+_)… (x+9)(x+9)

Page 9: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 10: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

x2-4x+4

GCF?– No.

Difference of squares?– No.

Factor. Factors of 1st & last terms x | 2,2; 1,4

Page 11: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

x2-4x+4 (cont’d)

(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)

Page 12: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 13: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2+6x+3

GCF?– 3

3(x2+2x+1) Factor Further?

– Yes.

Factor x2+2x+1 (x _ _)(x _ _)

Page 14: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2+6x+3 (cont’d)

+1 means?– 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)

Page 15: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 16: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

2x2+13x-7

GCF?– No.

Difference of squares?– No.

Factor… Factors of 2x2 are 2x,x

Page 17: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

2x2+13x-7 (cont’d)

(2x _ _)(x _ _) Factors of 7: 1,7 -7 means?

– One +, one -

+13x means?– Larger product must be +

Try it...

Page 18: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

2x2+13x-7 (cont’d)

(2x+1)(x-7)– Outer Product: -14x– Inner Product: +1x

• Sum: -13x

(2x-1)(x+7)– Outer Product: +14x– Inner Product: -1x

• Sum: +13x

Page 19: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

2x2+13x-7 (cont’d)

(2x+1)(x-7)– Outer Product: -14x– Inner Product: +1x

• Sum: -13x

*(2x-1)(x+7)* This is the one!– Outer Product: +14x– Inner Product: -1x

• Sum: +13x

Page 20: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 21: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

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

Page 22: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

4x3-36x (cont’d)

Roots: x, 3 4x(x _ 3)(x _ 3) 4x(x+3)(x-3) All done!

Page 23: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 24: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2-7x+1

GCF?– No.

Difference of squares?– No.

Factor?– 3x,x | 1,1

Try the various combinations...

Page 25: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2-7x+1 (cont’d)

(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.

Page 26: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10

Page 27: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2-11x+10

GCF?– No.

Difference of squares?– No.

Factors…– 3x, x | 1,10; 2,5

Try them...

Page 28: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2-11x+10 (cont’d)

(3x-1)(x-10)– Products: -30x, -x

• Sum: -31x

(3x-10)(x-1)– Products: -3x, -10x

• Sum: -13x

Page 29: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2-11x+10 (cont’d)

(3x-2)(x-5)– Products: -15x, -2x

• Sum: -17x

(3x-5)(x-2)– Products: -6x, -5x

• Sum: -11x

Page 30: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

3x2-11x+10 (cont’d)

(3x-2)(x-5)– Products: -15x, -2x

• Sum: -17x

*(3x-5)(x-2)* This is the one!– Products: -6x, -5x

• Sum: -11x

It works!

Page 31: The Factoring Puzzle 4x 3 +8x 2 +2xx 2 -64x 2 +18x+81 x 2 -4x+43x 2 +6x+32x 2 +13x-7 4x 3 -36x3x 2 -7x+13x 2 -11x+10.

The Factoring Puzzle

4x3+8x2+2x x2-64 x2+18x+81

x2-4x+4 3x2+6x+3 2x2+13x-7

4x3-36x 3x2-7x+1 3x2-11x+10


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