[Music] hello welcome to factoring this is one of my favorite topics pulling out the greatest common factor so going One Direction is factoring so when we have a B plus AC and we pull out the a that is called factoring when we distribute the a back through that is called distribute so most of the time you hear this property called the distributive property I like to call it the factoring property because that's where we miss it most of the time although that's what we're doing so for this first example what I want us to do is to find the greatest common factor so what is our greatest common factor what is common to both of these two terms it's 3x so when we pull 3x out of the first term what is left X what is left out of the second term is 2x so what you're doing to find out what is left is you're dividing the first term by 3x and you're dividing the second term by 3x to find what's left there and that's how you get the X Plus 2. next are formulas there are three that you use the most the first is difference of squares now it's listed by the factored terms first but it's named by the non-factored so it's a squared minus B squared and that factors into a minus B times a plus b this is probably the most common formula that you will ever see so an example of this is x squared minus 1 and that factors into x minus 1 times X Plus 1. again this is one of the most common factors most common polynomials that you'll ever see that you'll be asked to factor and it's also one of the most forgotten the next one is you need to realize that a plus a squared plus b squared is prime that means it cannot be factored a squared plus b squared cannot be factored well most people say well I can factor that that factors into a plus b times a plus b no it doesn't remember perfect squares a plus b times a plus b when you foil that out you get a squared plus two a B plus b squared you have that middle term so you have to remember that that middle term is always going to come up when you foil things that's why a squared plus b squared is prime so there's your perfect squares you have a plus or minus B quantity squared that becomes a squared plus or minus 2 a b and that last one is always plus b squared and your example is a plus 2 quantity squared becomes a squared plus 4X plus 4. sorry and I said a oops my bad all right now let's talk about how to factor something that looks like ax squared plus BX plus c students struggle with this so much they just they're like why can't I factor these nobody's ever taught me how to factor these well guys I'm going to try to teach you and break it down and make it simple the easiest way to do this is let's go back to multiplying so think about how do you multiply this if we multiply this we're going to go back to foiling that's where it all starts so how do you foil first outer inner last so my first are x squared my outer or minus 4X my inner are plus 3x my last or minus 12x simplified I get x squared minus x minus 12. now factoring is going the other direction so how do I take x squared minus x minus 12 to be factored yes I realize we have the answer at the top of the PowerPoint that's not the goal the goal is to teach you how to Think Through how to get back to that answer not oh the answer is at the top of the PowerPoint the goal again is How to Think Through to get that answer again we're going to use ax squared plus BX plus c c is 12. B is 1. we have signs in front of C which is minus sine in front of B which is minus okay here's my question what two factors of C which is 12 subtract that is the sign in front of 12 to give B which is one I'm going to repeat that what two factors of 12 will subtract to give me one here are my factors of 12. 1 times 12 2 times 6 3 times 4. 1 minus 12 is 11. I don't care about the sign that's not one two minus six is four that's not one three minus four is one bingo my answers my two factors are three and four so we found that three and four are the factors the larger gets the sign in front of B look back at my outer and inner the larger has to get the sign that is in front of B in order to get that sign in front of B so which is larger three or four four so that means that when I factor this I want x minus 4 times X Plus 3. it's that simple I wanted two factors of 12 that did the operation in front of 12 to get me the number in front of B the number one the larger of those two factors got the sign in front of the one it was that easy let's try another one for this one I'm going to talk through it before I go through the slides we want two factors of 15 to subtract to give us 2. the larger of the two factors will get a plus sign does everybody agree yeah yeah 15. our factors are 1 and 15 3 and 5. 1 minus 15 is 14 not that one 3 minus two is two oh it's going to be three and five which one is bigger three or five five so five is the larger one it gets the plus sign 3 gets the smaller it gets the minus how did I know one was plus and one was minus because the C in my a ax squared plus BX plus C was a minus that told me that my signs in my parentheses were different signs that one was positive and one was negative if that c had been a plus sign then both of my signs in my parentheses would be the same sign and they would be the sign in front of the B whoa that makes sense now let's multiply this out and make sure that we have the right answer you should never turn in a test paper where you haven't checked your answer if you're factoring you can multiply it out if you're solving an equation you can plug your answer in there are ways that you can always check your answers math is cool so to multiply this out I'm using the distributive property so I'm Distributing the X plus 5 through that second parentheses so X plus 5 times X and X plus 5 times minus 3. then the x times the X the X through the X plus 5 the minus 3 through the X plus five so I get x squared plus 5x minus 3x minus 15 so I get x squared plus 2X minus 15 which was my original yay okay oh no she put a number in front of the x squared not a problem what we're going to do to factor this one is we're going to multiply the 2 and the 3 to get six this time we want so think about our our first outer inner last to get our outer and our inner we multiply the 2 and the three when we do our outer and our inners so what are two factors of six that do the sign in front of the C to get me the seven so what's one plus six seven oh my gosh those are my my outer and my inner terms basically the one and the six are my outer and inner turns so using foil okay so my first is 2x times x my outer is 2x times three my inner is one times One X and my last is one times three x so that's what I mean by saying use think about how you would get your outer and inner so my first has to be 2X and X because I have to get that 2x squared and my last has to be 1 times 3x to get that 3. so now I need to break it up into my parentheses so I have 2X plus 1 and 1X Plus 3. so that leaves my answer as 2x plus 1 X Plus 3. and it wasn't as bad as you all thought now how would I factor this one completely first question you should ask yourself is do you have a greatest common factor well I see two what about an X so if I pull a 2X out then I'm left with x squared minus 4. is 4 a perfect square yes so then that factors using Ace the difference of squares a squared minus B squared so that factors into 2X times x minus 2 times X Plus 2. and that's my completed factored form what about this one no you changed it to the fourth power well is 16 a perfect square what about x to the fourth can I rewrite x to the fourth as x squared squared well that makes it a perfect square so then I'm left with using difference of squares to x squared plus 4 and x squared minus 4. now remember that x squared plus 4 is prime so I want to use the difference of squares again but I can only use it on the second term because the first term is prime so then I'm going to get x squared plus 4 times X plus 2 x minus 2. okay now last example how do I do this one so what if I group the greatest pull the greatest common factor out of the first two terms and the greatest common factor of the last two terms so in the first two terms I have 3x and the last two terms I have y so I have 3x times M squared minus N squared plus y times N squared minus N squared and that should be we're going to correct this real quick because nobody is perfect that should be an n now I'm going to factor out M squared minus N squared so if I grab that pull it really hard what's left is 3x plus y and then I'm going to factor M squared minus N squared using difference of squares and what I have is M minus n plus m plus n times 3x plus y and I'm done please let me know if you have any questions thank you [Music]