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Factoring
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  • 1. Factoring
  • 2. Factoring by Greatest Common Factor (GCF)
  • 3. Factoring by removing the GCF “undoes” the multiplication. (It’s the distributive property in reverse.) 2x2 – 4 2x2 – (2  2) 2(x2 – 2) Check 2(x2 – 2) 2x2 – 4
  • 4. Example 8x3 – 4x2 – 16x 2x2(4x) – x(4x) – 4(4x) 4x(2x2 – x – 4) 4x(2x2 – x – 4) 8x3 – 4x2 – 16x Check 
  • 5. Factoring ax2+bx+c
  • 6. 2 x bx c  To factor, find two numbers that: Add to b Multiply To c
  • 7. Example Factor x2 + 15x + 36. Check your answer. (x + )(x + ) (x + 1)(x + 36) = x2 + 37x + 36 (x + 2)(x + 18) = x2 + 20x + 36 (x + 3)(x + 12) = x2 + 15x + 36   
  • 8. Example Factor each trinomial. Check your answer. x2 + 10x + 24 (x + )(x + ) (x + 4)(x + 6) = x2 + 10x + 24
  • 9. Factor each trinomial. x2 + x – 20 (x + )(x + ) (x – 4)(x + 5)
  • 10. Remember to look for a GCF first! 3x2 + 18x – 21 3(x + )(x + ) 3(x – 1)(x + 7) 3(x2 + 6x – 7)
  • 11. Factor each trinomial. x2 – 11xy + 30y2 (x – 5y)(x – 6y)
  • 12. A trinomial is a perfect square if: • The first and last terms are perfect squares. • The middle term is double the product of the square roots. 9x2 + 12x + 4 3x 3x 2(3x 2) 2 2• •• REMEMBER THE X! Perfect Square Trinomials
  • 13. Factor. 81x2 + 90x + 25 Example (9x)(9x) (5)(5) The middle term = 2(9x)(5), so this is a perfect square trinomial
  • 14. Factor. Example 2 81 90 25 (9 )(9 ) x x x x    
  • 15. Factor. Example 2 81 90 25 (9 5 )(9 5) x x x x    
  • 16. A polynomial is a difference of two squares if: •There are two terms, one subtracted from the other. • Both terms are perfect squares. 4x2 – 9 2x 2x 3 3 For variables: All even powers are perfect squares Difference of Two Squares
  • 17. Example 2 49 100x  (7x)( 7x) (10)( 10)
  • 18. Example 2 49 100 (7 10)(7 10) x x x    Center term cancels out, so use opposite signs.
  • 19. Factoring ax2 + bx + c
  • 20. The “box” method 2 2x 6 First term Last term Include the signs!!! 2 2 7 6x x 
  • 21. Multiply a and c. (In this case, that would be 2 x 6 = 12) Put the factors of “ac” that add up to “b” in the other squares, with their signs and an “x”. Order does not matter. 2 2x 6 One factor The other factor 3x 4x 2 2 7 6x x  4 and 3 are factors of 12 that add up to 7, so they go in the empty spaces. Add an x because they represent the middle term.
  • 22. Factor the GCF out of each row or column. Use the signs from the closest term. 2 2x 63x 4xGCF is +2x
  • 23. Factor the GCF out of each row or column. Use the signs from the closest term. 2 2x 63x 4x+2x GCF is +3
  • 24. Factor the GCF out of each row or column. Use the signs from the closest term. 2 2x 63x 4x GCF is +x +2x +3
  • 25. Factor the GCF out of each row or column. Use the signs from the closest term. 2 2x 63x 4x +x GCF =+2 +2x +3
  • 26. The “outside” factors combine to factor the quadratic. 2 2x 63x 4x +x +2 +2x +3 (2x+3)(x+2)
  • 27. You can also use trial and error. Determine the possible factors for the first and last terms, and then keep trying combinations until you find the one that works. 2 3 17 10 x x First term: 3x and x are the only possible factors Last term: Factors are 1 and 10 or 2 and 5
  • 28. 2 (3 1)( 10) 3 31 10    x x x x 2 (3 10)( 1) 3 13 10    x x x x 2 (3 5)( 2) 3 11 10    x x x x 2 (3 2)( 5) 3 17 10    x x x x Only this combination works.
  • 29. Factoring by Grouping
  • 30. Factoring by grouping: If you have four terms – make 2 groups of 2 and factor out the GCF from each. MUST be used on 4 terms CAN be used on 3.
  • 31. Example Factor each polynomial by grouping. Check your answer. 6h4 – 4h3 + 12h – 8 (6h4 – 4h3) + (12h – 8) 2h3(3h – 2) + 4(3h – 2) 2h3(3h – 2) + 4(3h – 2) (3h – 2)(2h3 + 4)
  • 32. Factor each polynomial by grouping. Check your answer. Check (3h – 2)(2h3 + 4) 3h(2h3) + 3h(4) – 2(2h3) – 2(4) 6h4 + 12h – 4h3 – 8 6h4 – 4h3 + 12h – 8
  • 33. Example Factor each polynomial by grouping. Check your answer. 5y4 – 15y3 + y2 – 3y (5y4 – 15y3) + (y2 – 3y) 5y3(y – 3) + y(y – 3) 5y3(y – 3) + y(y – 3) (y – 3)(5y3 + y)
  • 34. Factor each polynomial by grouping. Check your answer. 5y4 – 15y3 + y2 – 3y Check (y – 3)(5y3 + y) y(5y3) + y(y) – 3(5y3) – 3(y) 5y4 + y2 – 15y3 – 3y 5y4 – 15y3 + y2 – 3y 
  • 35. You can use factoring by grouping on trinomials. 2 3 11 10x x  Split the 11x into two terms (coefficients should multiply to 30, because 3x10=30)
  • 36. 2 3 6 5 10x x x   Write the terms in whichever order will allow you to group.
  • 37. 2 (3 6 ) (5 10) 3 ( 2) 5( 2) (3 5)( 2) x x x x x x x x        