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The Factoring Puzzle

The Factoring Puzzle. 4x 3 +8x 2 +2x. GCF? Yes… 2x Factor it out… 2x(2x 2 +4x+1) Factor further? All done. The Factoring Puzzle. x 2 -64. GCF? No. Difference of Squares? Yes. Find the square roots of 1st & last terms (x+8)(x-8) More? No; all done. The Factoring Puzzle.

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The Factoring Puzzle

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  1. The Factoring Puzzle

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

  3. The Factoring Puzzle

  4. x2-64 • GCF? • 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

  6. x2+18x+81 • GCF? • No. • Difference of squares? • No. • 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

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

  11. 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)

  12. The Factoring Puzzle

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

  14. 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)

  15. The Factoring Puzzle

  16. 2x2+13x-7 • GCF? • No. • Difference of squares? • No. • 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) • 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) • 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

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

  23. The Factoring Puzzle

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

  25. 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.

  26. The Factoring Puzzle

  27. 3x2-11x+10 • GCF? • No. • Difference of squares? • No. • Factors… • 3x, x | 1,10; 2,5 • Try them...

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

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

  30. 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!

  31. The Factoring Puzzle

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