Factoring with the distributive property

what I want to do is start with an expression like 4x plus 18 and see if we can rewrite this as the product of two expressions is what essentially we're going to try to factor this and the key here is to figure out are there any common factors to both 4x and 18 and we can we can factor that common factor out we're essentially going to be reversing the distributive property so for example what is the largest number that is the order I could really say the largest expression that is divisible into both 4x and 18 well 4x is divisible by 2 because we know that 4 is divisible by 2 and 18 is also divisible by 2 so we can rewrite we can rewrite 4x as being as being 2 times 2x 2 times 2x if you multiply that so this is going to be 4x and then we can write 18 as the same thing as 2 times 9 2 times 9 and now it might be clear that you know when you apply the distributive property you will usually end up with a step that looks something like this now we're just going to undistribute the two right over Y we're going to factor the 2 out let me actually just draw that so we're going to factor we're going to factor the 2 out and so this is going to be 2 times 2x 2x plus 9 plus 9 and if you were to wanted to multiply this out it would be 2 times 2x plus 2 times 9 it would be exactly this which you would simplify as this right up here so there we have it we have written this as the product of 2 X 2 X press ins 2 times 2x plus 9 let's do this again so let's say that I have let's say that I have 12 plus 12 plus let me think of something interesting 12 plus 32 X 32 actually since we just to get a little bit of variety here let's put a Y here 12 plus 32 Y well what's the largest number that's divisible into both 12 and 32 2 is clearly divisible into both but so is 4 and let's see it doesn't look like anything larger than 4 is divisible into both 12 and 32 the greatest common factor of 12 and 32 is 4 and y is only divisible into the second term not into this first term right over here so it looks like 4 is the greatest common factor so we could rewrite each of these as a product of 4 and something else so for example 12 we can rewrite as we can rewrite as 4 times 3 and 32 we can rewrite so this is going to be plus 4 times 4 times well if you divide 32 Y by 4 it's going to be 8y and now once again we can factor out the 4 so this is going to be 4 times 3 3 plus 8 y 3 plus 8y and once you do more and more examples of this you're going to find that you can just kind of do the step all at once you could say what's the largest what's the largest number that's divisible into both of these what's 4 so let me factor 4 out 12 divided by 4 is 3 32 Y divided by 4 is 8 y