Rational Expressions Basic Introduction

in this video we're going to talk about rational expressions we're going to talk about how to simplify them how to multiply divide and add rational expressions so let's start with this one how can we simplify this particular rational expression well the first thing you want to do is you want to factor the expression completely and then cancel anything that you could cancel in the numerator we can factor out 7 because 35 and 7x are both divisible by seven if we divide 35 by seven we're gonna get five and if we divide negative seven by seven we're going to get negative now squared minus 25 what we have there is difference of perfect squares when you have the expression squared minus squared you can factor it as follows plus times minus the square root of squared is the square root of 25 is five so it's going to be plus five and minus five now is there anything that we can cancel right now it doesn't appear to be so but notice what happens if we factor out negative 1 from 5 minus if we take out negative the negative becomes positive and the positive five becomes negative five so now at this point we can cancel minus five doing so will give us our simplified answer which is negative seven over plus five so that's how we could simplify this particular rational expression now let's move on to our next example in this case we're multiplying two rational expressions how can we simplify this whole problem what we need to do is we need to factor everything let's begin with 4x minus 12. we could take out the gcf which is 4. 4x divided by 4 is negative 12 divided by 4 is negative 3. so that's the first thing we can do now we could do something similar with 6x plus thirty the gcf is six six divided by six is thirty divided by six is five now on the right we have trinomial with leading coefficient of one so to factor squared plus eight plus what we need to do is find two numbers that multiply to 15 but add to the middle coefficient eight so this is gonna be five and three five times three is fifteen five plus three is eight so we can write this as plus five times plus three and here we have difference of perfect squares so if we take the square root of squared we're going to get if we take the square root of nine we're going to get three one is going to be negative the other is going to be positive now that we factored everything what we need to do is cancel we could cancel these two factors plus three we can also cancel minus three and we can cancel plus five so what we're left over is four over six but now we can reduce this fraction four is two times two six is two times three so we can also cancel two thus the final answer for this problem is two over three now let's move on to our next problem that is divide in two rational expressions so how can we do this perhaps you heard of phrase called keep change flip it allows us to change from division problem to multiplication problem so what we're going to do is we're going to keep the first fraction we're going to change the sign from division to multiplication and we're going to flip the second fraction so let's rewrite the first fraction let's keep it the same we have squared plus 3x minus 10. divided by 3x squared plus 13x minus 10. now let's change the division sign to multiplication and then we're going to flip the second fraction so now we have problem that's similar to the last problem that we just solved so what we need to do at this point is factor and cancel so let's begin by factoring squared plus 3x minus 10. so we need two numbers that multiply to negative 10 but add to positive three what are those two numbers well we know five times two is ten but we need to use positive five and negative two because it adds up to positive three so this is going to be plus five times minus two i'm going to save this for later now let's focus on factoring that particular trinomial so we need two numbers that multiply to six but add to negative five so two times three is six but that adds up to positive five so we need to use negative two and negative three so this is going to be minus two times minus three now for this one what we need to do first is take out two because all of the coefficients are even so before we factor it we need to take out the gcf so dividing everything by two we're gonna have squared plus three minus eighteen so now we could focus on factoring this particular trinomial so we need two numbers that multiply to negative 18 but add to positive three so 18 is nine times two it's also six times three but if we use positive six and negative three this is going to work because these two add up to positive three so we still need the two in front but we could factor the trinomial highlighted in blue as plus 6 times minus 3. now let's delete few things to make extra space so what we need to do at this point is we need to factor that expression note that the leading coefficient is three it's not one and we can't factor out three from negative ten so we need to factor this special way what we're going to do is we're going to multiply the leading coefficient by the constant term 3 times negative 10 is negative 30. now we need two numbers that multiply to negative 30 but that's the middle coefficient 13. so if we divide negative 30 by 1 we'll get negative 30. if we divide it by 2 we'll get negative 15. if we divide it by 3 we'll get negative 10. 4 doesn't go into 30 but we could divide it by 5 and get negative 6. the one that is most promising is 2 and negative 15 because if we reverse the sign if we use positive 15 and negative 2 it now adds up to positive 13. so we're gonna rewrite this expression as three squared plus fifteen minus two minus ten so basically what did is replaced 13x with 15x and negative 2x because 15x minus 2x is still equal to 13x now let's factor out the gcf in the first two terms if we take out three we'll be left with plus five next let's take out the gcf in the last two terms if we take out negative two we'll be left with plus five so now we need to factor out plus five if we take out plus five from this term we're left with three if we take it away from this term we're left with negative two so that's we could factor this trinomial as follows it's going to be 3x minus 2 times plus 5. and you could foil it if you foil it you're going to get what you started with so now let's simplify our rational expressions so we could cancel something on top with something on the bottom in this case we can cancel the minus two factor we can also cancel the plus five factor and we could cancel minus three so what we're left over is two mean two times plus six divided by three minus two so this is the final answer now let's work on this problem so here we want to add two rational expressions how can we do this what we need to do is we need to get common denominators i'm going to multiply the second fraction by the other denominator top and bottom and then i'm going to multiply the first fraction by the denominator of the second fraction so right now have 2x times minus two divided by minus two times plus three and then plus five times plus three divided by minus 2 times plus 3. so what we can do at this point is we can write this as single fraction since we now have the same denominators so it's going to be 2x times minus 2 plus 5 times plus 3 divided by the denominator now the next thing we could do is simplify what we have on the numerator so let's distribute 2 times is two squared two times negative two is negative four and then we have five times and then five times three now we can combine like terms negative 4x plus 5x and that's going to be so we have 2x squared plus plus fifteen divided by minus two times plus three now this might be our final answer but we need to check to see if the trinomial on the numerator can be factored so let's multiply these two numbers 2 times 15 is 30. are there two numbers that multiply to positive 30 but add to the middle coefficient one well we know the factors of thirty one and thirty two and fifteen three and ten five and six none of these will add up to positive one in order for it to be positive thirty both numbers have to be either positive or both negative they have to have the same sign and so since this cannot be factored this is as far as we can go in terms of simplifying this expression so that's it for this video hopefully gave you good introduction in terms of the types of problems you'll see when studying rational expressions in algebra