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CCSS.Math: ,

you're probably familiar with the general term factor so if I were to say what are the factors of 12 you could say well what are the whole numbers that I can multiply by another whole number to get 12 so exit for examples you could say things like well I could multiply 1 times 12 to get 12 so you could say that 1 is a factor of 12 you could even say that 12 is a factor of 12 you could say 2 times 6 is equal to 12 so you could say that 2 is a factor of 12 and that 6 is also a factor of 12 and of course 3 times 4 is also 12 so both 3 & 4 are factors of 12 so if you said well what are the factors of 12 and you've seen this before you - well you could say 1 2 3 4 6 and 12 those are all factors of 12 you could also space it the other way around so let me just give an example so if I were to pick on 3 I could say that 3 is a factor factor of 12 or to phrase it slightly differently I could say that 12 is divisible divisible 12 is divisible by 3 now what I want to do in this video is extend this idea of being a factor or divisibility into the algebraic world so for example if I were to take 3 X Y so this is a monomial with an integer coefficient 3 is an integer right over here and if I were to multiply it with another monomial with an integer coefficient I don't know let's say times negative 2x squared Y to the 3rd power what is this going to be equal to well this would be equal to if we multiply the coefficients 3 times negative 2 is going to be negative 6 x times x squared is X to the third power and then Y times y to the third is y to the fourth power and so what we could say is if we wanted to say factors of negative 6 X to the 3rd Y to the 4th we could say that 3 X Y is a factor of this just as an example so let me write that down we could write that 3 X Y is a factor of is a factor of 6 of negative 6 X to the 3rd power y to the 4th or we could phrase that the other way around we could say that negative 6 X to the 3rd Y to the 4th is divisible by is divisible is divisible by 3 X Y so hopefully you're seeing the parallels if I'm taking these two monomials with integer coefficients that I multiply them and I get this other in this case this other monomial I could say that either one of these and there's actually other factors of this but I could say either one of these as a factor of this monomial or we could say that negative 6 X to the 3rd Y to the 4 is divisible by one of its factors and we could even extend this two binomials or polynomials for example if I were to take if I were to take let me scroll down a little bit whoops if I were to take let me say X plus 3 and I wanted to multiply it times X plus 7 we know that this is going to be equal to if I were to write it as a trinomial it's going to be x times X so x squared and then it's going to be 3x plus 7 X so plus 10 X and if any of this looks familiar we have a lot of videos where we go to detail of multiplying binomials like this and then 3 times 7 is 21 plus 21 and so because I've multiplied these two in this case binomials or we could consider themselves to be polynomials polynomials or binomials with integer coefficients notice the coefficient here they're 1 1 the constants here they're all integers because I'm dealing with all integers here we could say that either one of these binomials is a factor of this trinomial or we could say this trinomial is divisible by either one of these so let me write that down so I could say I'll just pick on X plus seven we could say that X plus 7 is a factor is a factor of x squared plus 10x plus 21 or we could say that x squared plus 10x plus 21 is divisible by is divisible by I could say X plus 3 or I could say X plus 7 is divisible by X plus 7 and the key is is that both of these binomials or even if we were dealing with polynomials we are dealing with things that have integer we're dealing with things that have integer coefficients