Math 25 Activity 6: Factoring Advanced


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1 Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult polynomials. We begin by reviewing a more complex GCF. 1. Find the greatest common factor of and, then factor. Your instructor will give you time to do this in a group and then explain on the whiteboard. Remember that when you factor, you find the GCF of all the terms, then divide each term by the GCF, then rewrite the expression in parentheses putting the GCF in front which indicates multiplication. 2. Find the greatest common factor of and, then factor. Your instructor will give you time to do this in a group and then explain on the whiteboard. In this problem, the GCF is actually a binomial itself. We continue with what is called factoring by grouping. Recall in a previous activity, we can group terms that are being added using the associative property of addition. When we have four terms, but none of them have a common factor, then we can try grouping them into pairs to see if that leads us to factoring the whole polynomial. Consider the following example explained step by step. Notice that the four terms do not have a common factor other than one. Look for a way to pair two terms together that have a common factor, then pair the last two terms together. The greatest common factor in the first grouping is 2, so factor the 2 from just the first two terms. The greatest common factor in the second grouping is u, so factor the u from just the second two terms. Notice how there are only two terms now and they are separated by a plus sign. Factor your new algebraic expression as you did in problem 2. Note: By the commutative property of multiplication, is the same as Page 1 of 5
2 Instructor! 3. Now as a group, try a similar example and write a short note explaining what you did in that step. We have given you the answer to let you check your work. Your instructor will give you time to do this in a group and then explain on the whiteboard. Try another example of factoring by grouping, except this time there are negative signs involved. Be careful when you factor out a negative number! 4. Try a similar example and write a short note explaining what you did in that step. We have given you the answer to let you check your work. 5. Factor yx 2y 4x + 8 Your instructor can explain the steps for these two problems on a whiteboard before continuing to the next skill. Page 2 of 5
3 Now that you have seen the skill that is called factor by grouping we can move to the last factoring skill we will go over in these activities. Note: There are MANY ways to factor trinomials. If you know how to factor trinomials quickly and accurately, then just practice factoring these examples your way. If you do not know how to factor, or you want to see another method, then practice these using this method. The goal is to factor a trinomial, which means we have three terms. We will do an example with each step explained, then you will spend the rest of the activity practicing these steps. You may need scratch paper for some of the inbetween steps. PROBLEM s STEPS STEP NOTES SCRATCH PAPER WORK Notice that the three terms do not have a common factor other than one. Label the coefficients so that the number before is the a coefficient, the number before is the b coefficient, and the number at the end is the c coefficient. Multiply List all the pairs that multiply to (from the previous step). Make sure to list them in order and stop listing before repeating yourself. Find the pair which adds to b. Since, we want to pick because = 11 Rewrite the middle term as the sum you found in the previous step. 3+8 = 11 so instead of write Make sure you have four terms now and pair them like you did in the factor by grouping problems. Find the GCF of each pair and factor each pair like you did in factor by grouping. Look for the GCF (which should be in the parentheses these should always match at this point) like you did with factor by grouping. Distribute to check your answer. Page 3 of 5
4 This example of what is called the AC Method of Factoring can work on most trinomials. It gets faster the more you practice it and it removes the guessing from factoring. 6. Try factoring the expression using the AC Method. Make sure you understand all the steps you wrote. Writing notes to yourself will help you follow your notes later. The answer should be. Students also need to think about negative signs. How can you tell when you need to add negative signs to a problem? Here are the four types of signs you can end up with, and what to do about the negative signs (this is why we list the pairs in order, without repeating): SITUATION EXAMPLE WHAT TO DO WHAT THE LIST LOOKS LIKE is positive multiply to, then all the positive is positive multiply to, then all the negative numbers are negative. is negative multiply to, then only the positive list of smaller numbers are negative and the list of larger is negative negative multiply to, then only the list of larger numbers are negative and the list of smaller Page 4 of 5
5 When it comes to factoring, practice and repetition really make a big difference. Students who practice factoring about 10 different trinomials each day will feel more comfortable with factoring, especially later in future math classes where factoring is an assumed skill necessary to finish more difficult problems. We do not have time to go over all the examples of factoring, but will leave lists of expressions that you can put on flashcards to practice factoring. Here is a formula for the special case of factoring a difference of squares: Some of these are more difficult than others. Trinomials: Binomials: Page 5 of 5
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