in this video we're going to talk about how to factor polynomials by taking out the GCF by using difference of perfect squares sum of perfect cubes different of cubes factoring trinomials using substitution factoring by grouping completing the square using synthetic division and also we're going to go over to some difficult problems at the end so let's start off the basics so let's say if you have this binomial 7x plus 21 how would you factor this expression now the first thing you should always look for is the GCF the greatest common factor the greatest common factor between 7 and 21 is 7 because 7 goes into itself and 21 so you want to factor out a 7 7 X / 7 if we write it out is X so that goes here 21 Y divided by this 7 gives you 3 y so that goes here and that's how you factor it by removing the GCF so let's try some other examples so go ahead and remove the GCF from 8 x squared + 12 X so Y squared and also from this one as well 36 X cubed + y squared minus 60 X to the fourth Y cube so what number goes into 8 and 12 for 4 goes into 8 and 12 and how many X variables can we remove from both terms on the left side we have 2 X variables on the right we have 1 so we can only remove one from each turn and the same is true for Y we can remove one Y from each term so if we take this term 8 x squared and divided by our GCF for X Y we're going to get 2x and if we divide 12 X Y squared by 4 X Y that's going to give us 3 y so that's what's left over to show your work you can do this 12 XY squared divided by 4 X Y notice that the X is canceled 12 divided by 4 is 3 y squared over Y you got to subtract the exponents 2 minus 1 you get Y to the 1 and that's what we put here so that's how we can factor that binomial by removing the GCF so let's try this one what is the greatest common factor between 36 and 16 12 goes into 36 and 60 and we can take out at least three X variables from both terms and we can take out two Y variables from each term so if we take this term and divided by 12 X cubed Y squared 12 divide 36 divided by 12 is 3 and we took out both all three X's in all two Y variables so there's nothing left over now for the next term negative 60 divided by 12 is negative 5 and out of the 4 X variables we remove 3 so this one left over and out of the three Y variables we only took out two so this 1y left over and that's how you factor it so how would you factor this expression x squared minus 25 notice that you can take the square root of x squared + 25 so this is a difference of perfect squares problem an equation that you need a squared minus B squared when you factor it it's going to be a plus B times a minus B so basically what you got to do is take the square root of x squared the square root of x squared is just X the square root of 25 is 5 and on one side you're going to have a plus and on the other side - so using that go ahead and factor y squared - 64 + 8 x squared minus 18 feel free to pause the video as you try these examples so the square root of Y squared is just Y and the square root of 64 is 8 and it's going to be plus and minus now we can't square root 8 or 18 but we could take out the GCF we can factor out a 2 8x squared divided by 2 is 4 x squared + 18 divided by 2 is 9 notice that 4 & 9 are perfect squares so now we can apply this equation so we're going to keep the 2 on the outside the square root of 4x squared is 2x and the square root of 9 is 3 and on one side we're going to have a positive side and on the other side and negative so that's the answer for that problem go ahead and try these examples 81 x squared - 36 y squared and try this one as well 200 X to the 4th minus 288 Y to the 6 so we can take the square root of 81 and 36 those are perfect squares so we can factor this one directly the square root of 81 x squared is 9x and the square root of 36 Y squared is 6 1 so we're going to have a plus and we're going to have a minus now notice that we can factor further notice that nine and six has a GCF which is three so in this term we can take out a 3 if we do so it's going to be a 3x plus 2y and we can take out a 3 from here as well and it's going to be 2x I mean 3x minus 2y so we could multiply by these two numbers so we have 9 times 3 x + 2 y + 3 x - 2 1 the fact that we have a 9 means that we could have removed a GCF let's say if we took out a 9 we would have 4 x squared 9 have 4x squared scuse me 9x squared 9x squared minus 4 y squared which would eventually factor to this expression so even if you don't take out the GCF at the beginning you can take it out at the end and you can still get the same answer now the square root of 9x squared is 3x and the square root of 4 y squared is 2y so both ways can work in the next problem we're going to factor out a 2 if we take out a 2 we're left with 100 which is a perfect square X to the 4 minus 144 Y to the 6 so let's go ahead and factor it using the difference of perfect squares equation so the square root of 100 X to the 4 if it's well the square root of 100 is 10 and the square root of x to the 4th you just divide the 4 by 2 you get x squared the square root of 144 is 12 and the square root of Y to the 6 divide the 6 by 2 you get 3 so 12 Y cubed and on one side it's + the other side of the - now 10 and 12 are both even numbers so we can take out a 2 and if we do so we'll be left with 5x squared plus 6y Q and we could take out another two from this term and so we'll have 5x squared minus 6y cubed so if we collect these streets numbers on the outside it's going to be see if I can fit it here 8 times 5x squared plus 6y q times 5x squared minus 6y cubed and that's the answer for this from so now how would you factor this trinomial x squared plus 11x plus 30 how would you do it now notice there's a 1 in front of the x squared the leading coefficient is 1 when that's the case all you need to do is look at the last term 30 and find two numbers that multiply to 30 but add to 11 let's start with 130 divided by 1 is 30 30 divided by 2 is 15 30 divided by 3 is 10 30 divided by 5 is 6 4 days ago into 30 now we need to pick a pair of numbers that add up to 11 5 plus 6 is 11 so the factory it's just going to be X plus 5 times X plus 6 by the way if you need to solve the equation let's say if this was equal to 0 and you want to find the zeros set each factor equal to 0 and solve for X so here X would equal negative 5 and X would equal negative 6 that's how you can find a zeros of a polynomial by factoring so let's try this one x squared plus 2x minus 15 so what are two numbers that multiply to negative 15 but add to the middle term 2 so let's divide negative 15 by 1 we'll get negative 15 it doesn't go into two but if we divide it by 3 we get negative 5 and if we divide by 5 we get negative 3 notice that 5 plus negative 3 is equal to positive 2 so the factor it's just X plus 5 times X minus 3 now what about this problem x squared minus 2x minus 48 let's look for two numbers that multiply to negative 48 but add to the middle turn negative 2 so if we divide negative 48 by 1 we're going to get negative 48 if we divide it by 2 it's going to give us negative 24 if we divide it by 3 negative 16 by 4 negative 12 5 doesn't go into negative 48 if we divided by 6 negative 8 if we divided by 8 and negative 6 at that point the numbers will repeat in reverse order so we need two numbers that add up to negative 2 6 plus negative 8 is negative 2 so the answer is X plus 6 and X minus 8 if was equal to 0 the solution to the equation will be negative 6 and positive 8 you got to change the sign so let's try this one x squared minus 9x plus 20 so let's look for two numbers that multiply to 20 but add to negative 9 if we divide 20 by one week we're going to get 20 if we divided by 2 we'll get ten 3 doesn't go into 20 but if we divided by 4 we'll get 5 now 4 times 5 is 20 but 4 plus 5 adds up to 9 negative 4 and negative 5 also multiplies to positive 20 but they add to negative 9 so the answer is X minus 4 times X minus 5 so that's how you can factor a trinomial when the leading coefficient is 1 now what can we do when the leading coefficient is not 1 so let's say if you want to factor 2x squared minus 5x minus 3 now notice that the on the leading coefficient is a small number it's 2 so sometimes it might be easier to factor it by trial and error other times you can use a method some people call it the Athey method but for this one let's factor it by trial and error because the numbers are small so we need two numbers that multiply to 2x squared the only way to make this work is 2x and X it has to be that way now we need two numbers that multiply to negative 3 so to get 3 in the first place we need a 1 and a 3 the question is should it be like this so 1 3 or should it be 3 1 right now let's let's try 3 1 if we multiply 2x by one we'll get 2x and 3 times X is 3x now the only way you can add up to negative 5x is if we have negative 2x and negative 3x however to get negative 3 we need a positive and a negative number so that's not going to work so we're going to change the order we're going to put the 3 here and the 1 here 2 times 3 is 6 X 1 times X is 1x now 6 plus 1 is 7 but 6 minus 1 is 5 now this could work since you want a negative 5 we need a negative 3 and the plus 1 because 2x times negative 3 is negative 6x + 1 plus X is 1x that adds to negative 5x and that's how you can factor it using them trial and error now let's say if you don't like that method if you want a consistent method to follow here's what you could do let's use the same equation we're going to multiply the first and the last coefficient together so 2 times negative 3 is negative 6 and we need two numbers that multiply to negative 6 but add to the middle term negative 5 so if we divide it by 1 we're going to get negative 6 if we divide it by 2 we'll get negative 3 1 plus negative 6 is negative 5 so we're going to be placed negative 5 X with positive 1 X and negative 6 X because X minus 6 is negative 5 X in the first two terms we're going to factor out the GCF the greatest common factor between 2x squared and X it's just X - x squared divided by X is 2x + 1 X divided by X is 1 now we're going to take out the GCF between the last two terms we're going to factor out negative 3 negative 6x divided by negative 3 is 2x negative 3 divided by negative 3 is plus 1 if these two are the same then you're on the right track so we're going to factor out 2x + 1 now this whole thing is one turn if we take out the 2x plus one we're left with X which goes here and if we take out the 2x plus one from this term will left with negative three which goes here and that's our answer 2x plus 1 and X minus 3 so let's try some more examples using that method so feel free to try this problem 3x squared plus 5x minus 2 so let's multiply the first and last coefficient so 3 times negative 2 is negative 6 and if we divide negative 6 by 1 we're going to get negative 6 if we divided by 2 we'll get negative 3 if we divide it by 3 we'll get negative 2 and if we divide by 6 we'll get negative 1 so which pair of numbers add up to 5 6 plus negative 1 is equal to 5 so what we're going to do is replace 6 I mean 5 X with 6 X minus 1 X and that let's factor by grouping so between 3 x squared + 6 X the GCF is 3 X 3 x squared divided by 3x is X 6x divided by 3 X is 2 now for the last two terms we're going to take out a negative 1 negative 1 X divided by negative 1 is X negative 2 divided by negative 1 is plus 2 so here we have a common factor we're going to factor out X plus 2 so if we take out X plus 2 we're left with 3x and if we remove X plus 2 here we're left with negative 1 and that's how we factor it so let's try one final example on this topic let's try 6x squared minus 7x minus 3 so let's multiply 6 and negative 3 6 times negative 3 is negative 18 and if we divide it by 1 we're going to get negative 18 if we divide it by 2 we'll get negative 9 2 plus negative 9 adds up to the middle term negative 7 so we have is 6x squared plus 2x minus 9 X minus 3 in the first two terms we're going to take out the GCF which is 2x 6x squared divided by 2x is 3x 2 X divided by 2x is 1 in the last two terms we're going to take out a negative 3 negative 9x divided by negative 3 is negative 3x and negative 3 divided by negative 3x plus 1 so here is our common factor we're going to factor it out and if we take out 3x plus 1 here we're left with 2x and if we remove 3x plus 1 from the second term we'll left with negative 3 and so that's how we can factor a trinomial when the leading coefficient is not 1 so our next topic we're going to talk about how to factor polynomials with three terms by using substitution so let's say if you have X to the fourth plus 7x squared plus 12 so notice that this exponent is twice the value of that one when you see that you can factor it just as any trinomial as we did before where the leading coefficient is one so let's substitute it with a variable let's say a is equal to whatever the middle term is x squared just without the coefficient so therefore a squared is X to the fourth so we have a squared plus seven a plus 12 so we need two numbers that multiply to 12 add to 7 12 divided by one is 12 12 divided by 2 is 6 12 divided by 3 is 4 but 3 plus 4 is 7 so this is going to be a plus 3 times a plus 4 which is if we replace a with x squared it's x squared plus 3 and x squared plus 4 so that's the answer for that example let's try this one x2 this X to the six actuate hold on negative two X to the 6 plus 6 X cubed plus 56 now notice that we have even numbers here so what we can do is take out the GCF which is negative 2 we'll be left with x squared minus 3x cubed minus 28 so now we're going to focus on this part here but let's substitute the middle term with a so we're going to say a is X cubed which means that a squared is X to the 6 so what we now have is negative 2 a squared minus 3 a minus 28 so we need two numbers that multiply to negative 28 but adds to the middle term negative 3 so if we divide it by one we'll get negative 28 if we divide it by two we we're going to get negative 14 3 doesn't go into 28 but four does negative 28 divided by four is negative seven and four plus negative seven is negative three so the factor it's going to be a plus 4 and a minus seven so we're going to be placed a with X cubed so it's going to be X cubed plus 4 and X cubed minus seven try this one X is to the negative two plus 13x to the minus one plus 4t and this one we're going to solve it as well in addition to factor in this particular function so we're going to set a equal to the middle term which is X to the minus one so right now we have a squared plus 13 a plus 4t so we need two numbers that multiply to 40 but that add to 13 40 divided by one is 40 divided by 2 is 20 divided by 4 is 10 divided by 8 5 is 8 and 8 plus 5 is 13 so it's a plus 5 a plus 8 and at this point we can replace a with X to the minus 1 plus 5 and X to the minus 1 plus 8 so if we set each factor equal to zero X to the negative 1 is equal to negative 5 and X to the minus 1 is equal to negative 8 now to solve for X need to understand that X to the negative 1 is the same as 1 over X so 1 over X is equal to negative 5 which we can write it as a negative 5 over 1 and we can cross multiply so 1 is equal to negative 5 X dividing both sides by negative 5 our answer is negative 1 over 5 so for the other one if we follow the same process X will be negative 1 over 8 so that's how you solve a particular function like that so what about this one let's say if e to the 2x if we have e to the 2x plus 10 e to the X plus 21 how would you factor it notice that the exponent here is twice the value of the middle one so this is another substitution problem so we're going to set a equal to this middle term e to the X so a squared would be e to the 2x so what two numbers multiply to 21 but add to the middle turn 10 this would have to be 3 & 7 3 times 7 is 21 but 3 plus 7 is 10 so right now what we have is a squared plus 10 a plus 21 and when we factor it's going to be a plus 3 times a plus 7 and then you need to replace a with e to the X so it's e to the X plus 3 times e to the X plus 7 and that's how you factor so our next topic they need to know is the sum and difference of perfect cubes so let's say if you want to factor X cubed plus 8 the equation that you need is 8 to the third plus the B to the third is equal to a plus B times a squared minus a B plus B squared so 8 to the third is X to the third and B to the third is 8 so to find a you need to take the cube root of x to the third the cube root of x to the third is X you just divide the three by three and the cube root of 8 is 2 because 2 times 2 times 2 is 8 so a squared if a is if a is X a squared therefore is x squared and then we have minus a times B or x times 2 which is just 2x and B is 2 so B squared or 2 squared is 4 and that's how you factor it so let's say if you want to factor Y Q minus 125 this is a difference of perfect cubes so if there's a minus here this is going to be minus and this is going to be plus everything else is the same the cube root of Y cube is just Y and the cube root of 125 is 5 because 5 times 5 times 5 3 times is 125 and because we have a minus sign it's going to stay negative see these two signs stay the same but it switches here so a squared is y squared and to find a B it's going to be plus we change the sign from negative to positive plus 5 times y which five y and then B squared 5 squared is 25 let's try another example so let's say if you want to factor 27 X cubed plus 64 Y to the sixth so the cube root of 27 is 3 and the cube root of x to the third is X the cube root of 64 since we have a plus this is going to be plus the cube root of 64 is 4 4 times 4 times 4 64 and the cube root of Y to the 6 you just got to divide this by 3 it's going to be Y squared so keep in mind this is a and this is B so a squared 3x times 3x will therefore be 9x squared and then minus a be 3x times 4y squared is 12 X Y squared and then B squared so 4y squared times 4 y squared is 16 y to the fourth and that's how you factor it you just got to follow this equation you can solve for a and B and just substitute it in the equation so now let's say if we have this cubic polynomial and we want to factor by grouping now we already use this method when we factored using the AC method when the trinomial had a leading coefficient that wasn't one but here's something you need to know notice that the first two coefficients one and two has the same ratio as negative five negative ten two divided by one is 2 negative 10 divided by negative five is two when the first two coefficients have the same make sure as the last two coefficients you can therefore factor by grouping if that ratio doesn't if the ratio is not the same it won't work so in the first two terms let's take out the GCF which is x squared and so x cubed divided by x squared is x two x squared divided by x squared is two and in the last two terms let's take out a negative five so we'll be left with X plus two so here's our common factor we're going to take our X plus two and then we'll be left with x squared minus five and that's how you do it let's try another one so 4x cubed minus 8x squared plus 3x minus six so negative 8 divided by 4 is negative 2 and negative 6 divided by 3 is also negative 2 so they have the same ratio so which means we can factor by grouping in the first two terms let's take out the GCF which is 4x squared and so we'll be left with X minus 2 and in the last two terms let's take out a 2 actually not a 2 by 3 and so we'll also have X minus 2 so our answer is going to be X minus 2 times 4x squared plus 3 so how would you factor x squared minus 4 and x squared plus 4 how would you do it now we covered the first one already that's the difference of perfect squares we know it's going to be X plus 2 and X minus 2 but what about x squared plus 4 a sum of perfect squares can we factor that the answer is no we can't factor it using real numbers however we could factor it using imaginary numbers x squared plus 4 is X plus 2i and X minus 2i I is equal to the square root of negative 1 if you haven't learned imaginary numbers you can skip this section and move on to the next topic but I'm going to foil it x times X is x squared x times negative 2i is negative 2i x 2i times X is plus 2i X and 2 I times negative 2i is negative 4i squared so the two middle terms cancel and you get x squared minus 4i squared now notice that I is the square root of negative 1 I squared is equal to negative 1 so we now have is x squared minus 4 times negative 1 which is x squared plus 4 so that's how you can factor the sum of perfect cubes using imaginary numbers so let's try some examples including the use of radicals let's say if you want to factor x squared plus 9 x squared minus 3 and x squared plus 3 so when you have the sum of perfect cubes you're going to use imaginary numbers but I'm going to compare it to x squared minus 9 x squared minus 9 is going to be X plus 3 X minus 3 but x squared plus 9 it's going to be X plus 3i and X minus 3i so all you're going to do is add an eye to it now if you want to factor x squared minus 3 here's we need to do it's the difference of perfect squares so the square root of x squared is X and the square root of 3 is actually radical 3 so we have plus and minus now here we have x squared plus 3 so it's just X plus two root three times I and X minus root three times I so depending on what level of course you're taking you may or may not need to know that so now we're going to focus on factoring higher degree polynomials using synthetic division so let's say if you have x cubed minus 2x squared minus 5x plus 6 notice that 1 & 2 doesn't have the same ratio as 5 & 6 that we can't factor by grouping so we need a different method so what you want to do is make a list of your possible factors possible factors of 6 are 1 2 3 & 6 and possible factors of the leading coefficient 1 is just plus minus 1 so all of these are possible answers let's start with positive 1 let's see if that works so we're going to use synthetic division so the coefficients that you see 1 negative 2 negative 5 & 6 we're going to place them here and bring down the 1 in synthetic division you're going to multiply add multiply add and just repeat the process so 1 times 1 is 1 and then negative 2 plus 1 is negative 1 and then this one times negative 1 is negative 1 negative 5 plus negative 1 is negative 6 1 times negative 6 is negative 6 6 2 negative 6 is 0 so usually when you use synthetic division this goes down by one level if this was X to the 3rd this one's going to be x squared so we have 1 x squared minus 1 X minus 6 now we can use synthetic division again but right now we have a trinomial we know how to factor trinomial so two numbers that multiply to negative 6 but add to the middle term negative 1 would be let's see we have 1 and negative 6 to a negative 3 and this is what we're looking for so this trinomial can be factored 2 X plus 2 times X minus 3 however our first solution was positive 1 and if and it's factored form it is X minus one so this right here is how you factor the original expression if you multiply these three factors you're going to get this original equation by the way if you get a zero that means this number works it's a factor if this wasn't a zero you have to pick a different number these are the possible factors as you can see 2 negative 3 and negative 1 they're all in possibilities here let's try another one like this so let's say if we have X to the fourth plus 2x cubed plus x squared plus 8x minus 12 so possible factors of 12 are 1 2 3 4 6 and I'm out of space for 12 as well and possible factors of the lien coefficient 1 is just 1 so let's see what numbers are going to work for this one let's try 1 again let's see what happens so let's put the coefficients 1 2 1 8 negative 12 by the way if one of these variables were missing let's save those if there was no x squared if it was like 2x cubed plus 8x make sure you put a zero in place for the x squared don't forget that that's important so let's bring down the 1 1 times 1 is 1 2 plus 1 is 3 1 times 3 is 3 1 plus 3 is 4 1 times 4 is 4 8 plus 4 is 12 1 times 12 is 12 those two add to 0 so right now we have this was 4 so now it's going to go down to 3 and 1 X cubed plus 3x squared plus 4x plus 12 now notice that the coefficients 1 & 3 have the same ratio as 4 and 12 12 divided by 4 is 3 3 divided by 1 is 3 which means we can factor by grouping so in the first two terms let's take out the GCF we know it's x squared and we'll be left with X plus three and the last two terms let's factor out a four and we're going to get X plus three so the answer is x squared plus 4 times X plus 3 but that's just for this part now keep in mind we have a 0 here so it's factored form is X minus 1 so this is how we can completely factor the problem using real factors now if you want to include imaginary numbers you can factor the x squared plus 4 which is the sum of perfect squares and if you want to factor it it's going to be X plus 2i X minus 2i and then times X plus 3 so the next thing we're going to do is factor trinomials by completing the square so let's say if you have x squared plus 6x plus 7 now we're going to ignore the 7 at least for now look at the 6x look at the coefficient 6 take half of 6 half of 6 is 3 and then squared now we still have the plus 7 since we added 9 we change the value of the expression which is not good so to balance that we need to take away 9 so if we add 9 and subtract 9 we haven't changed the value of the expression and that's what that's what we want to do now this term is x squared plus 6x plus 9 two numbers that multiply to 9 but add to 6 is 3 and 3 so it can be factored as X plus 3 X plus 3 this is a perfect square because the 3 is are the same but you really don't need to use this method there's a shortcut in which you can use to factor that perfect square and his we need to do so whatever this variable is is going to go here X whatever this sign is it's going to go there plus and whatever this number is before you square it so not 9 but 3 it's going to go here and this 2 is going to go here as well and does this can be written as X plus 3 squared because there's two of them and then these two numbers we simply need to combine seven minus nine is negative two and that's how you can factor by completing the square so let's try another example so let's say if we have x squared minus 4x plus 12 now that method is going to be extremely useful when you have fractions because you don't want to factor a fraction so let's take 1/2 of 4 1/2 of 4 is ignore the negative sign half of 4 is 2 and we're going to square it and since we're adding 2 squared which has the value of 4 we need to subtract 4 so that the expression maintains its original value so the fact of this expression it's going to be X minus 2 squared and 12 minus 4 is 8 that's it alright let's try this example x squared minus 3x plus 1 so we have x squared minus 3x half of 3 is 3 over 2 and we got a square it so plus 1 now 3 over 2 squared 3 squared is 9 2 squared is 4 since we added 9 fourth we have to take away 9 fourths so the factor it's going to be X minus 3 over 2 squared now to combine 1 & 9 over 4 we'll need to get common denominators 4 over 4 is the same as 1 so 4 over 4 minus 9 over 4 is negative 5 over 4 so this is our final answer X minus 3 over 2 squared minus 5/4 so let's try this one 2x squared plus 6x minus 9 so first we need to factor out a 2 so we'll be left with x squared plus 3x and half of 3 is 3 over 2 and we're going to square it and then we have minus 9 so we've added 3 over 2 squared so I'm going to take away 3 over 2 squared but notice that the 3 over 2 squared is distributed by this 2 so we need to incorporate that 2 there so we're going to leave the to only outside and then it's going to be X plus three over two squared and then we have minus 9 minus 3 squared is 9 2 squared is 4 times 2 so this part is going to stay the same we don't have to worry about it anymore and that's a terrible 3 I did it again 9 over 4 times 2 if we divide 4 by 2 backwards we get 2 so we're just going to get a 2 on the bottom and now we need to get common denominators so we're going to multiply this by 2 over 2 so this is going to be negative 18 divided by 2 minus 9 over 2 and negative 18 minus 9 is negative 27 over 2 and we still have our original expression so this is the answer that's how we can factor by completing the square so that's enough with that topic so now let's factor some advanced factoring polynomial problems so let's say if you get x squared minus 2xy plus y squared minus 9 what would you do here when there's so many variables how would you factor this expression now feel free to pause the video and try it out yourself it helps to see if you focus on this part first now you need to know some equations a plus B squared is a plus B times a plus B and when you foil it it becomes a squared plus 2 a B plus B squared a minus B squared is a squared minus 2 a B plus B squared notice that this equation is the same as what we see here instead of a and B is just x and y so you can factor those to be as X minus y squared and we still have a negative not now notice that we have a difference of perfect squares like x squared minus 25 which when we factor it we know it's X plus 5 X minus 5 so we have that same situation here but I'm going to use brackets this time if we take the square root of x minus 5 squared the square goes away and we're just left with X minus y in both parentheses I mean brackets and if we take the square root of 9 it's 3 so we have X minus y plus 3 X minus y minus 3 and that's how you factor it try this one x squared y squared minus y squared minus Z squared plus x squared Z squared how would you factor that expression notice that the first two terms have a Y squared and the last two terms have a Z squared now notice that we have a minus y squared here it's like positive then negative but here is negative then positive I'm going to be arrange it so that it's in the same order positive and then negative and then the second part I'm going to write it as x squared Z squared minus Z squared because notice that these two are similar and these two are similar the coefficient is 1 and negative 1 1 and negative 1 they have the same ratio which means we can probably factor by grouping so in the first two terms let's take out the GCF which is y squared so we're going to be left with x squared minus 1 and in the last two terms let's take out the GCF which is Z squared and we're going to be left with x squared minus 1 so we have a common factor x squared minus 1 if we take that out here we're going to have Y squared and here we're going to have plus Z squared so the factor x squared minus 1 is going to be X plus 1 times X minus 1 and then Y squared plus x squared so that's the answer for this problem and if you want imaginary solutions I guess you can say why plus Zi and Y minus Zi but I don't think that's necessary for this problem but that concludes this video so we covered almost every factoring technique shortcut trick that you'll probably need as you go through your algebra and trigonometry and precalculus courses and even calculus you can take this information and it's going to help you along your way so thanks for watching this video and have a great day