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

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Presentation transcript:

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

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

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

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.

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

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

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+_)

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

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

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

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)

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

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

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)

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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!

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