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factorization

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factorization

  1. 1. Factoring Polynomials 1032 −− xx This process is basically the REVERSE of the distributive property. =−+ )5)(2( xx distributive property
  2. 2. =−− 1032 xx In factoring you start with a polynomial (2 or more terms) and you want to rewrite it as a product (or as a single term) Factoring Polynomials )5)(2( −+ xx Three terms One term
  3. 3. What is a perfect square?
  4. 4. 652 ++ xx We need to find factors of 6 ….that add up to 5
  5. 5. Factoring Trinomials, continued... 652 ++ xx 2 x 3 = 6 2 + 3 = 5 Use the numbers 2 and 3 to factor the trinomial… Write the parenthesis, with An “x” in front of each. ( )3)2( ++ xxWrite in the two numbers we found above. ( )xx )(
  6. 6. 652 ++ xx You can check your work by multiplying back to get the original answer ( )3)2( ++ xx ( )3)2( ++= xx =+++= 6232 xxx 652 ++= xx So we factored the trinomial… Factoring Trinomials, continued...
  7. 7. Factoring Trinomials 61 65 67 2 2 2 −+ −− ++ xx xx xx Find factors of – 6 that add up to –5 Find factors of 6 that add up to 7 Find factors of – 6 that add up to 1 6 and 1 – 6 and 1 3 and –2
  8. 8. 61 65 67 2 2 2 −+ −− ++ xx xx xx factors of 6 that add up to 7: 6 and 1 ( )1)6( ++ xx factors of – 6 that add up to – 5: – 6 and 1 factors of – 6 that add up to 1: 3 and – 2 ( )1)6( +− xx ( )2)3( −+ xx Factoring Trinomials
  9. 9. Factoring Trinomials The hard case – “Box Method” 62 2 −+ xx Note: The coefficient of x2 is different from 1. In this case it is 2 62 2 −+ xx First: Multiply 2 and –6: 2 (– 6) = – 12 1 Next: Find factors of – 12 that add up to 1 – 3 and 4
  10. 10. Factoring Trinomials The hard case – “Box Method” 62 2 −+ xx 1. Draw a 2 by 2 grid. 2. Write the first term in the upper left-hand corner 3. Write the last term in the lower right-hand corner. 2 2x 6−
  11. 11. Factoring Trinomials The hard case – “Box Method” 62 2 −+ xx – 3 x 4 = – 12 – 3 + 4 = 1 1. Take the two numbers –3 and 4, and put them, complete with signs and variables, in the diagonal corners, like this: 2 2x 6− It does not matter which way you do the diagonal entries! Find factors of – 12 that add up to 1 –3 x 4x
  12. 12. The hard case – “Box Method” 1. Then factor like this: 2 2x 6− x3− x4 Factor Top Row Factor Bottom Row 2 2 2x 6− x3− x4 x From Left Column From Right Column 2 2x 6− x3− x42 x x2 2 2x 6− x3− x4 x2 2 x 3− x
  13. 13. The hard case – “Box Method” 2 2x 6− x3− x4 x2 2 x 3− )32)(2(62 2 −+=−+ xxxx Note: The signs for the bottom row entry and the right column entry come from the closest term that you are factoring from. DO NOT FORGET THE SIGNS!! ++ Now that we have factored our box we can read off our answer:
  14. 14. The hard case – “Box Method” 2 4x 12 x16− x3− x 3 x4 4 =+− 12194 2 xx Finally, you can check your work by multiplying back to get the original answer. Look for factors of 48 that add up to –19 – 16 and – 3 )4)(34(12194 2 −−=+− xxxx
  15. 15. Use “Box” method to factor the following trinomials. 1. 2x2 + 7x + 3 2. 4x2 – 8x – 21 3. 2x2 – x – 6
  16. 16. Factoring the Difference of Two Squares The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared. a2 – b2 = (a + b)(a – b)FORMULA: (a + b)(a – b) = a2 – ab + ab – b2 = a2 – b2
  17. 17. Factoring the difference of two squares Factor x2 – 4y2 Factor 16r2 – 25 (x)2 (2y) 2 (x – 2y)(x + 2y) Now you can check the results… (4r) 2 (5) 2 Difference of two squares Difference Of two squares (4r – 5)(4r + 5) a2 – b2 = (a + b)(a – b)
  18. 18. The information was taken from the following people on slideshare: Estela, Sep 22 2013. Factorising quadratic expressions 1http://www.slideshare.net/estelav/factorising-quadratic-expressions-1 Julia Li,http://www.slideshare.net/jagheterjuliali/ch-06-10762231 Majapamaya, Nov 13, 2013. 05 perfect square, difference of two squareshttp://www.slideshare.net/majapamaya/05-perfect-square-difference-of-two-squares Swart J.E, Oct 28 2013. Factoring and Box Methodhttp://www.slideshare.net/swartzje/factoring-and-box-method

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