The Factoring Puzzle 4 x 38 x 22

x 2 -64 4 GCF? – No. 4 Difference of Squares? – Yes. 4

x 2+18 x+81 4 GCF? – No. 4 Difference of squares? – No. 4

x 2+18 x+81 (cont’d) 4 X; 1, 81; 9, 9; 4, 27; … 4

x 2+18 x+81 (cont’d) 4 Then what? 4 (x+_)… 4 (x+9)

x 2 -4 x+4 4 GCF? – No. 4 Difference of squares? – No.

x 2 -4 x+4 (cont’d) 4 (x_ _) 4 +4 means? – Both factors

3 x 2+6 x+3 4 GCF? – 3 4 3(x 2+2 x+1) 4 Factor

3 x 2+6 x+3 (cont’d) 4 +1 means? – Both signs are the same

2 x 2+13 x-7 4 GCF? – No. 4 Difference of squares? – No.

2 x 2+13 x-7 (cont’d) 4 (2 x _ _)(x _ _) 4 Factors

2 x 2+13 x-7 (cont’d) 4 (2 x+1)(x-7) – Outer Product: -14 x –

4 x 3 -36 x (cont’d) 4 Roots: x, 3 4 4 x(x _

3 x 2 -7 x+1 4 GCF? – No. 4 Difference of squares? –

3 x 2 -7 x+1 (cont’d) 4 (3 x+1)(x-1) – Products: -3 x, +1

3 x 2 -11 x+10 4 GCF? – No. 4 Difference of squares? –

3 x 2 -11 x+10 (cont’d) 4 (3 x-1)(x-10) – Products: -30 x, -x

3 x 2 -11 x+10 (cont’d) 4 (3 x-2)(x-5) – Products: -15 x, -2

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The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

4 x 3+8 x 2+2 x 4 GCF? – Yes… • 2 x 4 Factor it out… – 2 x(2 x 2+4 x+1) 4 Factor further? – All done.

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

x 2 -64 4 GCF? – No. 4 Difference of Squares? – Yes. 4 Find the square roots of 1 st & last terms 4 (x+8)(x-8) 4 More? – No; all done.

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

x 2+18 x+81 4 GCF? – No. 4 Difference of squares? – No. 4 Factor? 4 What are the factors of the 1 st & last terms?

x 2+18 x+81 (cont’d) 4 X; 1, 81; 9, 9; 4, 27; … 4 (x _ _) 4 +81 means? – Both the same sign 4 +18 x means? – Both must be + 4 (x+_)

x 2+18 x+81 (cont’d) 4 Then what? 4 (x+_)… 4 (x+9)

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

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

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

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

3 x 2+6 x+3 4 GCF? – 3 4 3(x 2+2 x+1) 4 Factor Further? – Yes. 4 Factor x 2+2 x+1 4 (x _ _)

3 x 2+6 x+3 (cont’d) 4 +1 means? – Both signs are the same 4 +2 x means? – Both factors must be sums (+) 4 Find the factors… 4 (x + 1) 4 Final Answer: 3(x+1)

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

2 x 2+13 x-7 4 GCF? – No. 4 Difference of squares? – No. 4 Factor… 4 Factors of 2 x 2 are 2 x, x

2 x 2+13 x-7 (cont’d) 4 (2 x _ _)(x _ _) 4 Factors of 7: 1, 7 4 -7 means? – One +, one 4 +13 x means? – Larger product must be + 4 Try it. . .

2 x 2+13 x-7 (cont’d) 4 (2 x+1)(x-7) – Outer Product: -14 x – Inner Product: +1 x • Sum: -13 x 4 (2 x-1)(x+7) – Outer Product: +14 x – Inner Product: -1 x • Sum: +13 x

2 x 2+13 x-7 (cont’d) 4 (2 x+1)(x-7) – Outer Product: -14 x – Inner Product: +1 x • Sum: -13 x 4 *(2 x-1)(x+7)* This is the one! – Outer Product: +14 x – Inner Product: -1 x • Sum: +13 x

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

4 x 3 -36 x 4 GCF? – 4 x 4 4 x(x 2 -9) 4 Factor Further? – Yes; is (x 2 -9) a difference of squares? • Yes 4 Find the square roots of the 1 st & last terms

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

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

3 x 2 -7 x+1 4 GCF? – No. 4 Difference of squares? – No. 4 Factor? – 3 x, x | 1, 1 4 Try the various combinations. . .

3 x 2 -7 x+1 (cont’d) 4 (3 x+1)(x-1) – Products: -3 x, +1 x • Sum: -2 x 4 (3 x-1)(x+1) – Products: +3 x, -1 x • Sum: +2 x 4 No matches, therefore 3 x 2 -7 x+1 is prime.

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10

3 x 2 -11 x+10 4 GCF? – No. 4 Difference of squares? – No. 4 Factors… – 3 x, x | 1, 10; 2, 5 4 Try them. . .

3 x 2 -11 x+10 (cont’d) 4 (3 x-1)(x-10) – Products: -30 x, -x • Sum: -31 x 4 (3 x-10)(x-1) – Products: -3 x, -10 x • Sum: -13 x

3 x 2 -11 x+10 (cont’d) 4 (3 x-2)(x-5) – Products: -15 x, -2 x • Sum: -17 x 4 (3 x-5)(x-2) – Products: -6 x, -5 x • Sum: -11 x

3 x 2 -11 x+10 (cont’d) 4 (3 x-2)(x-5) – Products: -15 x, -2 x • Sum: -17 x 4 *(3 x-5)(x-2)* This is the one! – Products: -6 x, -5 x • Sum: -11 x 4 It works!

The Factoring Puzzle 4 x 3+8 x 2+2 x x 2 -64 x 2+18 x+81 x 2 -4 x+4 3 x 2+6 x+3 2 x 2+13 x-7 4 x 3 -36 x 3 x 2 -7 x+1 3 x 2 -11 x+10