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we're asked to apply the distributive property and we have one-half times the expression 2a minus 6b plus 8 so to figure this out I've actually already copied and pasted this problem onto my scratch pad I have it right over here 1/2 times 2a minus 6b plus 8 so let me just rewrite it so I'm going to take it only color-coded - just for fun so it's going to be 1/2 times get myself some space 1/2 times 2a minus 6b so 2a minus 6b minus 6 write it this way minus 6b and then we have plus 8 plus and I will do 8 in this color plus 8 and so I just need to distribute the 1/2 if I'm multiplying one half times this entire expression that means I multiply 1/2 times each of these terms so I'm going to multiply 1/2 times this 1/2 times this and 1/2 times that so 1/2 times 2 way so it's going to be 1/2 times 2a times let me do that same color so you see where the 2a came from 1/2 times 2a minus minus 1/2 times 6 B minus 1/2 times 6 B x 6b plus plus 1/2 times 8 1/2 x times 8 and so what's this going to be well let's see I have 1/2 times 2a 1/2 times 2 is just 1 so you're just going to be left with a and then you have minus 1/2 times 6 B well we could just think about what 1/2 times 6 is going to be 1/2 times 6 is going to be 3 and then you still are multiplying behind B there's going to be 3 B and then we have plus 1/2 times 8 half of 8 is 4 or do you say 8 halves is equal to 4 wholes all right so this is going to be 4 so it's a minus 3b plus 4a minus 3b plus four so let's type that in it's going to be a minus 3b plus four and unless I just it's it's literally half of each of these terms half of 2a is a half of 6 B's 3b so we have minus 6b so is going to be minus 3b and then plus 8 instead of that half of that plus 4 so let's check our answer and we got it right let's do let's do another one of these so let's say so they say apply the distributive property to factor out the greatest common factor and here we have 6 DM minus 4t so let me get my scratch pad out again so I'm running out of space that way so we have alright like this we have 60 60 M minus 40 minus 40 so what is the greatest common factor of 60 M + 40 well 10 might jump out at us we might say okay look you know 60 is 10 so we could say this is the same thing as 10 times 6 and actually and then of course you have the M there so you could use this 10 times 6m and then you could view you could view this as 10 times 4 but we 10 still isn't the greatest common factor and you say well how do you know that well because 4 & 6 still share a factor in common they still share - so what if you're actually factoring out the greatest common factor what's left should not share a factor with each other so let me let me think even harder about what a greatest common factor of 60 and 40 is well 2 times 10 is 20 so you could actually factor out a 20 so you have 20 and 30 M sorry Oh 20 and 3m and 40 can be factored out into 20 and 20 and 2 and now 3m and - 3m and to share no common factor so you know that you have fully factored these two things out if you think that this is something kind of a strange art that I just did one way to think about greatest common factors you say okay 60 you can literally do a prime factorization you save 60 is 2 times 30 which is 2 times 15 which is 3 times 5 so that's 60 s prime factorization 2 times 2 times 3 times 5 and then 40 s prime factorization is 2 times 20 20 is 2 times 10 10 is 2 times 5 so that right over here this is 40 s prime factorization and to get out the greatest common factor you want to get out as many common prime factors so you have here you have two 2s and a 5 here you have two 2s and a 5 you can't go to three twos into five because there aren't three 2s and a 5 over here so you have two twos and a 5 here two twos and a 5 here so two times two times five is going to be the greatest common factor so 2 times 2 times 5 that's 4 times 5 4 times 5 that is 20 that's one way of kind of very systematically figuring out a greatest common factor but anyway now that we know that 20 is the greatest common factor let's factor it out so this is going to be equal to 20 times so 60 divided 60m divided by 20 you're just going to be left with 3m just going to be left with 3m and then minus minus 40 divided by 20 you're just left with the 2 minus 2 minus 2 so let's type that in so this is going to be 20 times 20 times 3 M 3 M minus 2 and once again we feel good that we literally we did take out the greatest common factor because 3m and 2 especially 3 and 2 are now relatively prime relatively prime just means they don't share any factors in common other than 1