Algebra 1 Advanced U8 D9 Notes

Hinzugefügt: 2026-05-12

[00:00:01]hey everyone welcome into Unit 8 day n notes um today we're going to focus on factoring quadratic trinomials when a equals 1 and Prime and GCF so Our intention today is that students will be able to factor quadratic trinomials when a equals 1 with a GCF and identify Prime trinomials with 80% accuracy okay so yes we discussed the steps of factoring when a equals 1 today we're going to go into a few additional things um and not necessarily steps but things we need to consider when we are factoring quadratic trinomials so we still have the standard form that we are going to be uh looking and fact looking at these quadratic trinomials and factoring them where we have ax^2 plus BX plus C where A and B are coefficients of those terms and then we have C is the constant so again below are some of the things to consider when factoring and the first thing we need to consider when factoring is to always always always check for a GCF first so you're going to notice every single example we have in these notes begins with GCF and we write the GCF if there's a GCF or we write no if there is none or no GCF the second thing we need to consider is ask ourselves the question can the expression Factor again so we can factor with GCF we can also Factor like we did yesterday in day eight notes when we're looking for factors of C that are going to add to B and then the final step or thing to consider is if neither of the above can happen then it is prime we can't fact Factor it at all again I want to emphasize when neither of those two happen so not if one of them happens or the other it's if neither of the above happen we write prime meaning we can't Factor it okay so let's take a look at some examples number one h^2 + 11 H + 24 so the first thing we do check for a GCF if we look at each term there is no number that goes into 1 11 and 24 other than one so nothing there and then um only two of the terms have the variable H so therefore there is no GCF okay then we move into our steps like we did yesterday if there's no GCF we move on and we say okay we have factors of positive 24 that we're looking for so we list them out 1 24 2 12 3 8 and 4 six again we have to make sure we consider all of the factors so -1 -4 -2 -12 -38 and then you guessed it4 -6 go ahead we find the SU of each of these and we're looking for which of the factors are going to add to 11 so 1 and 24 makes 25 so that's not it 2 and 12 is 14 that's not it 3 and 8 that's 11 so that is the one that we are considering and looking for and again talk about strategies if we know that the factors we need be to be both positive or both negative and the number we need to add to is a positive then it is impossible for these factors to add to a positive number so it's just a quick shortcut a tip or a trick um to help speed up the process of factoring quadratic trinomials and in this case we only consider the factors on the left because those are the ones that have the possibility of adding to a positive number whereas the ones on the right um only have the possibility of adding to a negative number okay so moving forward we need to finish this process so we have oops we're using the letter H so we have H+ three and then we have H+ 8 and there we have it we have factored the CU in this part is that we are looking for a GCF and there was no GCF so that's kind of the one of the new things that we are considering when we are factoring all right let's go on to number two so again we are always always always looking for a GCF first and in number two again there's going to be no GCF and that's okay we're going to uh continue to write that out and consider it just so that we don't forget when there is a GCF okay now we're going to go ahead and look for for factors of 11 so factors of 11 we have 1 and 11 and then we have1 11 now we're looking to see okay which of these factors add to5 1 + 11 is 12 1+1 is -12 so neither of our factors add to that middle term that um coefficient that we call B so we can't Factor it that way and there's no GCF so no GCF or ability to factor and therefore it is prime okay so this is the situation that we consider a prime when we can't find a GCF and then we cannot Factor all right let's go ahead and jump all the way down to number five so for number five again we're looking for a GCF and this time we have found a GCF the GCF is we consider we see that every single term can be divisible by two but in this case if we remember our first term a we do not want it to be negative so we are going to divide out and factor out a negative2 additionally if we look at every single term they each share the variable Z and then we look for the smallest exponent out of all them which is z to the first power so that's our GCF so -2 Z -2 Z -2 to Z okay before we do our dividing we remember our GCF comes along for the ride on the outside of our set of parenthesis here and now we Factor so -2 -2 those - tws cancel out and we are left with Z the 3 divid Z the 1st which we subtract the exponents 3 minus one which gives us Z to the second power then we go ahead do 20 / -2 which gives us a -10 and then Z to the 2 power over Z to the first power is just Z to the first Power and then we have POS 36id -2 is -8 and then those Z's will cancel out so this is what we are left with okay now we want to see can we Factor this again so we're looking for factors of8 and so here we go let's list them out we have -1 * 18 -2 let's rewrite that -2 * 9 -3 * 6 then we flip it 1-8 scoot it over a little bit 1 8 2 9 and then 3 -6 all right let's add these up so we have 17 7 3 -7 -7 and -3 okay so now we look do any of these um factors set of factors add up to -10 and in this case none of them add up to -10 so then this is just our final answer it is not prime because we factored with a GCF remember if neither of the options happen then it's Prime but if we can do at least one of them it's not prime so I just maybe note that not prime okay all right uh go ahead take a few minutes to try the other examples come back and we'll see how we did see you all in a few all right welcome back let's go ahead and check out the answers to 3 4 and six so if we go ahead and check out for number three first our GCF is negative 1 because even though there's you know it's not definitive that you know we have a solid number like we did in number five it was like -2 there still is a negative one there and we do not want that first term to be negative right the first term is negative one so we automatically then divide out by a negative one because a negative divide out by a negative will make it positive and we don't want that first term being negative so we divide up by negative one that's going to flip all of our signs um don't forget to write our um GCF out in front and then when we go ahead and look through our factors um we come up with the factors of 2 and 12 and again we're looking for two positive factors because after we divide out by negative 1 it's positive4 four and we're trying to get to a positive 14 so the only way we can get to a positive 14 with adding is by adding to positive factors so we know that we're not going to use any of the negative factors in this situation so if you have that mental note down and that little trick then you can know when you're looking at your factors you should only be writing out and looking at the positive factors jumping over to number four very similar situation um we do have a GCF of four so after we divide out the four we go ahead and we get c^2 - 9 C + 14 so once again our term C the number we're looking for factors is positive and the number we're looking to add to is 9 so in this case positive factors of a positive number can be positive times a positive or A negative times a negative in this case we are not interested in the positive times a positive numbers because those two positive numbers can only add to a positive number we are interested in the negative factors because those are the ones that could add to a -9 and we find out that the factors are -2 and -7 remember don't forget to write that GCF in front right here and then the last one we'll look at is number six so there's no GCF and that doesn't mean it's Prime because we can go ahead and Factor it and when we Factor it we come to the conclusion conclusion that the factors that add to -1 are 4 and5 awesome great job today y'all um if you have any questions make sure you reach out to your teacher otherwise have a great rest of your day peace out