Intro to rational expression simplification

when we first started learning about fractions or rational numbers we learned about the idea of putting things in lowest terms so if we saw something like 3 6 we knew that 3 and 6 share a common factor we know that the numerator well 3 is just 3 but that 6 could be written as 2 times 3 and say since they share a common factor the 3 in this case we could divide the numerator by 3 and the denominator by 3 or we could say that this is just 3 over 3 and they would cancel out and in lowest terms this fraction would be one half or just to kind of hit the point home if we had eight over eight over twenty-four once again we know that this is the same thing as eight over three times eight or this is the same thing as one over three times eight over eight the eights cancel out and we get this in lowest terms as one third the same exact idea applies to rational expressions these are rational numbers rational expressions are essentially the same thing but instead of the numerator being an actual number and the denominator being an actual number they're expressions involving variables so let me show you what i'm talking about let's say i had let's say that i had 9x 9x plus 3 over over 12 x plus 4. now this numerator up here we can factor it we can factor out a three this is equal to three times three x plus one that's what our numerator is equal to and our denominator we can factor out a four this is the same thing as 4 times 3x 3x 12 divided by 4 is 3 12x over 4 is 3x plus 4 divided by 4 is 1. so here just like there the numerator and the denominator have a common factor in this case it's three x plus one in this case it's a rational it's a it's a variable expression it's not an actual number but we can do the exact same thing they cancel out so if we were to write this rational expression in lowest terms we could say that this is equal to 3 over 4 that's equal to 3 over 4. let's do another one let's do another one let's say that we had let's say that we had x squared let me see a good one so let's say we had x squared x squared minus 9 over 5 x plus 15 over 5 x plus 15. so what is this going to be equal to what is this going to be equal to so the numerator we can factor it's a difference of squares we have x plus three times x minus three and in the denominator we can factor a five out this is five times x plus three so once again a common factor in the numerator and in the denominator we can cancel them out but we touched on this a couple of videos ago we have to be very careful we can cancel them out we can say that this is going to be equal to x minus 3 over 5 but we have to exclude the the values of x that would have made this denominator equal to zero that would have made the entire expression undefined so we could write this as being equal to x minus three over five but x cannot be equal to negative three negative three would make this zero or it would make this whole thing zero so this and this whole thing are equivalent this is not equivalent to this right here because this is defined at x is equal to negative 3 while this isn't defined at x is equal to negative 3. so to make them the same i also have to add the the extra the extra condition that x cannot equal negative 3. and so likewise over here if this was a function if this was let's say this was we wrote y is equal to 9x plus 3 over 12x plus 4 and we wanted to graph it when we simplify it the temptation is well you know we factored out a 3x plus 1 in the numerator and the denominator they cancel out the temptations say well this is the same graph as y is equal to the constant 3 4 which is just a horizontal line at y is equal to 3 4. but we have to add one condition we have to eliminate we have to exclude the x values that would have made this thing right here equal to zero and that would have been zero if x is equal to negative one third if x is equal to negative one third this or this denominator would be equal to zero so even over here we'd have to say x cannot be equal to negative one third that condition is what really makes is what really makes this is what really makes that equal to that that x cannot be equal to negative one-third let's do a couple more of these and i'll do these in pink let's say that i had let's say that i had x squared x squared plus six x plus eight over over x squared plus 4x actually even better let me let me do this a little bit x squared plus 6x plus 5 over over x squared minus x minus 2. so once again we want to factor the numerator and denominator just like we did with traditional numbers when we first learned factoring when we first learned about fractions and lowest terms so if we factor the numerator what two numbers when i multiply them equal 5 and i add them equal 6. well the numbers that pop into my head are 5 and 1. so the numerator is x plus 5 times x minus 1 and then our denominator two numbers multiply negative 2 adam negative 1. negative 2 and positive 1 pop out of my head negative oh this is a positive 1 right x plus 5 times x plus 1 right 1 times 5 is 5. 5x plus 1x is 6x so here we have negative we have a a positive one and a negative two so x minus two times x plus one times x plus one so we have a common factor in the numerator and the denominator these cancel out so you could say that this is equal to x plus 5 over x minus 2 but for them to really be equal we have to add the condition we have to add the condition that x cannot be equal to negative one we cannot because if x is equal to negative one this is undefined we have to add that condition because this by itself is defined at x is equal to negative one you could put negative 1 here and you're going to get a number but this is not defined at x is equal to negative 1. so we have to add this condition for this to truly be equal to that