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“Bottoms Up”

Factoring ax 2 + bx + c. “Bottoms Up”. Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is 6 Leading term is 6 Therefore 6 * 6 is 36. 6 x 2 + 13x + 6.

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“Bottoms Up”

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  1. Factoring ax2 + bx + c “Bottoms Up”

  2. Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is 6 Leading term is 6 Therefore 6 * 6 is 36 6x2 + 13x + 6

  3. Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. x2 + 13x + 36 6x2 + 13x + 6

  4. Step 3: Factor the new equation from step 2: x2 + 13x +36 (x + 4)(x + 9)

  5. Step 4: Divide each constant term in the factored form by the original coefficient of the leading term. Original coefficient of the leading term: 6 (x + 4)(x + 9)

  6. Then reduce the fractions:

  7. Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:

  8. Check it- use FOIL:

  9. Remember you may check some equations using graphing calculator- type in the equation in y = and find where it crosses the x-axis. Not all equations cross the x-axis. This particular one crosses at x = -2/3 and x = -3/2. Look at these solutions and the end result of step 4, and then look at step 5. Factoring leads to solutions, the zeros, the roots.

  10. Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is -3 Leading term is 10 Therefore -3 * 10 is -30 10x2 - x - 3

  11. Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. x2 - x - 30 10x2 - x - 3

  12. Step 3: Factor the new equation from step 2: x2 - x - 30 (x + 5)(x - 6) x2 - x - 30

  13. Step 4: Divide each constant term in the factored form by the original coefficient of the leading term. Original coefficient of the leading term: 10 (x + 5)(x - 6) (x + 5)(x - 6) 10 10

  14. Then reduce the fractions: (x + 1)(x - 3) 2 5

  15. Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get: (2x + 1)(5x – 3) (x + 1)(x - 3) 2 5

  16. Check it- use FOIL: 10x2 -6x + 5x – 3 10x2 - x – 3 (2x + 1)(5x – 3)

  17. 6x2 - 2x – 28 Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is -28 Leading term is 6 Therefore 6 * -28 is -168 You Try!

  18. Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. x2 -2x -168 6x2 - 2x – 28

  19. Step 3: Factor the new equation x2 -2x -168 ( x -14 )(x +12 )

  20. Step 4: Divide each constant term in the factored form by the original coefficient of the leading term. Original coefficient of the leading term: 6 ( x -14 )(x +12 ) 6 6 ( x -14 )(x +12 )

  21. Then reduce the fractions: ( x-7 )(x +2 ) 3 1 Step 5: Now we use the “bottoms up” ( x -14 )(x +12 ) 6 6

  22. ( 3x-7 )(x +2 ) Check your answer with factoring (your choice). Conversation with a caution 6x2 - 2x – 28 Purpose of factoring

  23. Step 1: multiply (c) (a), Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient. Step 3: Factor the new equation Step 4: Divide each constant term in the factored form by the original coefficient of the leading term and then reduce the fractions: Step 5: Now we use the “bottoms up” Quick steps!

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