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Factoring Quadratics: Difficult Trinomial

Factoring Quadratics: Difficult Trinomial. ( a, b, and c have no common factors.). Method of DECOMPOSITION. A Difficult Trinomial fits the format given by ax 2 +bx+c where a, b, and c have no Common Factors and a ≠1 .

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Factoring Quadratics: Difficult Trinomial

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  1. Factoring Quadratics:Difficult Trinomial (a, b, and c have no common factors.) Method of DECOMPOSITION

  2. A Difficult Trinomial fits the format given by ax2+bx+c where a, b, and c have no Common Factors and a ≠1.

  3. The Method of DECOMPOSITION for factoring a Difficult Trinomial, (ax2+bx+c where a, b, and c have no common factors and a ≠1), breaks down (decomposes) the middle term so there are now four terms and the Grouping* method of factoring can be used. * If review is needed, search the OERB for Factoring Quadratics-Grouping .

  4. To factor a DifficultTrinomial, using DECOMPOSITION, the following general rule is used: where pxq = acand p+q = b and the new expression on the right must be factored using Grouping.

  5. (1)Set up the product and sum needed here. To factor using DECOMPOSTION:

  6. (2) List the needed factors until the usable factors are found. Remember to insert any needed negative signs.

  7. (3) Fill the slots with the usable factors to check that both conditions are met. -3 10 -3 10

  8. (4) Set up the DECOMPOSTION with a=6, p=-3 and q=10 : -3 10 -3 10

  9. (5) Factor the expression using Grouping*. *If needed, check the OERB for Factoring Quadratics-Grouping to review Grouping.

  10. This completes the factoring using DECOMPOSITION.

  11. Example 1: Factor fully using DECOMPOSITION.

  12. Solution: (1) Set up the product and sum needed here.

  13. Solution: (2) List the needed factors until the usable factors are found. Remember to insert any needed negative signs.

  14. Solution: (3) Fill the slots with the usable factors to check that both conditions are met.

  15. -3 4 Solution: -3 4 (4) Set up the DECOMPOSITION with a=3, p=-3 and q=4 :

  16. -3 4 -3 4 Solution: (5) Factor the expression using Grouping.

  17. -3 4 -3 4 Solution: This completes the factoring.

  18. Now try part b).

  19. Solution cont’d: • Set up the product and sum needed here. • List the needed factors until the usable factors are found. Remember to insert any needed negative signs. • Fill the slots with the usable factors to check that both conditions are met. • Set up the DECOMPOSITION. • Factor the expression using Grouping.

  20. -3 -8 -3 -8 Solution cont’d: Here’s the solution.

  21. It’s time to try part c).

  22. Solution: Follow the steps outlined in the previous slides, noting there is a Common Factor that should be removed first.

  23. -1 10 Solution cont’d: Here’s one solution. -1 10

  24. -1 10 Solution cont’d: Here’s an alternate solution if the Common Factor was not removed first. -1 10

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