Date Lesson #6: Factoring Trinomials with Leading Coefficients. Day #1

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1 Algebra I Module 3: Quadratic Functions Lessons 6-7 Name Period Date Lesson #6: Factoring Trinomials with Leading Coefficients Day #1 New week, new challenges! Last week, we reviewed how to factor using GCFs, learned how to factor by grouping, and worked on factoring basic trinomials. It s time to step up our factoring game here. Example #1 Factor x 2 5x 14 But what happens when we are asked to factor something like 2x x + 5? How is this problem different from Example #1? Are there a pair of numbers that multiply to 5 but add/subtract to 11? Whenever you have a problem where there is a leading coefficient other than 1, you need to be extra careful when you factor. Example #2 Factor 2x x + 20 STEPS 1.) Multiply the first and last numbers 2.) What 2 numbers multiply to 40, but add up to 13? 3.) Rewrite 13x as the sum or difference of 8x and 5x 4.) Find the GCF of the first 2 terms and the last 2 terms 5.) Factor out what binomial is in common 6.) Write the leftovers as a binomial 7.) Multiply to check your answer

2 When you break down the middle term, does it matter which number comes first? Let s look. Example #3 Factor 3x x + 6 a c = 3x 2 + 9x + 2x + 6 3x 2 + 2x + 9x + 6 Example #4 2x x + 8 a c = Example #5 2x x + 15 a c = Example #6 2x 2 + 9x + 4 a c =

3 Try these 1.) 3x 2 + 8x ) 3x x ) 4x x ) 2x x ) 3x x ) 2x 2 + 7x ) 4x x ) 2x x + 14

4 Day #2 So we spent yesterday focusing on some advanced factoring strategies. Hopefully by now, you ve gotten a bit better with them. Let s step it up. What happens when there is both positive and negatives involved? Yikes! It is basically the same process as we ve been dealing with but I will leave you with this bit of advice to make your life easier. If you have a positive and a negative, it is often easiest to make sure the negative comes first Example #1 Factor 2x 2 7x 15 a c = Example #2 Factor 3x 2 22x 16 a c = Example #3 Factor 2x 2 5x 25 a c =

5 Example #4 Factor 3x x 14 a c = Try these 1.) Factor 3x 2 + 5x 12 2.) Factor 2x x 30 3.) Factor 2x 2 7x 30 4.) Factor 3x x 20 Challenge Question 5.) Factor 4x 2 4x 15 6.) Factor 6x 2 5x - 21

6 Day #3 Ok, now we are going to wrap up our work with this type of factoring. We just talked about when we have both positives and negatives. We talked about how it is helpful to have the negative come first, whenever it is possible. But what if it s not possible? What if the negative comes second? What if BOTH signs are negative? WHAT ON EARTH ARE WE GOING TO DO?!?! Example #1 Factor 2x 2 11x + 12 a c = Example #2 Factor 4x 2 11x + 6 a c = Example #3 Factor 5x 2 21x + 4 a c = Example #4 Factor 6x 2 11x + 3 a c =

7 Try these 1.) Factor 8x 2 34x ) Factor 6x 2 19x ) Factor 2x 2 11x ) Factor 3x 2 23x + 30 Challenge Question 5.) Factor 6x 2 25x ) Factor 6x 2 35x + 36

8 Lesson #7: Factoring Completely Ok. We just spent a LOT of time on factoring. We talked about how to factor by using GCFs. We learned how to factor the difference of two perfect squares. We also discussed factoring trinomials. For the next two days, we are going to work on factoring completely. It s nothing necessarily NEW it basically means you re going to have to factor the same expression more than once. When you are asked to factor completely, it is a giveaway that you re going to have to factor more than once. Now we have to look at the leftovers and see if we can further break that down too. STEP #1: STEP #2: STEP #3: STEP #4: Ask yourself, Is there a GCF I can pull out? If so, factor out the GCF. Look at the leftovers and ask yourself, Is this DOTS? If so, factor by using the difference of two squares. If it is not DOTS, is it a trinomial? If so, try breaking it down using the trinomial method. It is possible that after you factor a problem once, you are done but if it says factor completely, you are pretty much guaranteed you will need to factor it more than one time. Completely factor the following expressions 1.) 2x x ) x 3 + 2x 2 35x 3.) 3x ) 4x 3 + 8x 2 32x

9 5.) -x 2 13x 40 6.) -x 2 2x ) -2x 2 4x ) -3x x 2 48x CLASSWORK: Completely factor the following expressions 1.) 2x x ) 3x 2 21x ) x 3 100x 4.) 5x x x 5.) 3x 4 18x x 2 6.) 2x 5 2x 4 40x 3 7.) -x 2 12x 20 8.) -2x 2 4x ) -3x x 2 120x

10 Day #2 Today we are going to continue with factoring expressions completely. It is the same general idea as yesterday; We are going to break down these expressions more than once. The problems we dealt with yesterday all contained either a GCF or a negative (or both) that we needed to factor out before using DOTS or trinomial factoring. Today we are going to focus on factoring completely when there isn t a GCF. Completely factor the following expressions 1.) x 4 + 2x ) x 4 2x ) x 4 5x ) x 4 6x ) x ) 16x ) x ) 81x 8 1

11 CLASSWORK: Completely factor the following expressions 1.) x 4 5x ) x 4 13x ) x 4 + 5x ) x 4 + x ) x ) 1 x 4 7.) x 4 + x ) 16x ) x 4 7x ) x 4 4x ) x 4 14x ) x 4 81