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Transcript
Math 60 5.2: Factoring by Grouping Elementary Algebra In order to understand Factoring by Grouping it helps to review some aspects of polynomial multiplication. Multiply by using the Distributive Property Multiply by using the FOIL Method x 32 x 1 x2 x 1 32 x 1 x 32 x 1 2 x 2 x 6 x 3 2x 2 x 6x 3 F O I L 2 x 5x 3 2 2 x 2 5x 3 F O I L Now we will be taking these multiply problems and reversing the procedure to produce a reverse distributive process called Factoring by Grouping. Factoring by Grouping is used exclusively to factor 4–termed polynomials, where a 4–termed polynomial expression is rewritten as a product of two binomial factors. Factor by Grouping: Many students consider this a "reverse distributive process". Practice Problems: Factor by Grouping. You can check your work on any factoring problem by taking your final answer and multiplying back using an appropriate technique (see §4.5) to verify the original polynomial. 2 x2 x 6 x 3 2 x2 x 6x 3 2 2 x x 6 x 3 x 2 x 1 3 2 x 1 2 x 1 x 3 x 3 2 x 1 For “Factoring by Grouping” problems the easiest way to check your work is by FOIL-ing. For the last two terms of the 4–termed polynomial… Lead on a NEGATIVE Factor out a NEGATIVE. Lead on a POSITIVE Factor out a POSITIVE. 1. x 2 x 3x 3 2. 9w 2 6w 6 w 4 3. 12 x2 42 x 10 x 35 4. x2 3xy 4 xy 12 y 2 5. y3 3 y 2 y 2 6 6. y3 3 y 2 y 2 6 (put in descending order first) Whenever you are asked to factor polynomials the expectation is that this process is done completely. In other words, when we write our factorization we need to verify that each polynomial in the factorization is non-factorable (or prime). Sometimes it may be necessary to factor more than once to produce this “complete” factorization. Factor completely by Grouping. 7. 3z 4 3z 3 7 z 3 7 z 2 3z 4 3z 3 7 z 3 7 z 2 8. 9 x 3 6 x 2 45x 2 30 x 9 x 3 6 x 2 45x 2 30 x Sometimes the terms in a 4–termed polynomial are not in the proper order to make Factoring by Grouping possible. If the 4–termed polynomial is factorable, then it is simply a matter of producing the correct arrangement of terms, so that we can then use Factoring by Grouping. Not all arrangements will allow for Factoring by Grouping; however, of those that work, the answer will all be algebraically equivalent regardless of the arrangement of terms. (Hint: Review Problems 5 and 6 on the previous page.) Factor by Grouping. 9. ax 10 5x 2a Answers: 1. 4. 7. 10. ac bd ad bc 10. For Chapter 5: Where appropriate the POSITIVE factor has been listed first. Remember, that since multiplication is commutative: x 3 x 4 x 4 x 3 x 3 x 1 x 4 y x 3 y z 2 z 1 3z 7 x b c d 2. 5. 8. 3w 2 y 2 y 2 3 3x 3x 2 x 5 2 3. 6. 9. 2 x 7 6 x 5 y 2 y 2 3 x 2 a 5