in this video we're gonna talk about factoring how to factor the greatest common factor the GCF so let's say if we have the expression 3x plus 15 what is the greatest common factor between 3x + 15 3x + 15 are both divisible by what number they're both divisible by 3 so therefore 3 is the GCF the greatest common factor 3x divided by 3 is simply x + 15 divided by 3 is 5 and so that's how you can factor an expression by removing the GCF the greatest common factor so that's the answer for the first one now let's try some other examples try this one 7x minus 28 go ahead and factor this binomial by removing the GCF so 7x and negative 28 are both divisible by 7 7x divided by 7 is X negative 28 divided by 7 is negative 4 and so that's the answer now go ahead and try these two problems 4x squared plus 8x and also 5x squared minus 15 X cubed so 4x squared plus 8x they're divisible by what they're both divisible by what term both are divisible by 4 and they're both divisible by X so 4x is the GCF 4x squared divided by 4x is simply X 8x divided by 4x is 2 and so that's the solution for the first one now for the next one 5x squared + 15 X Q are both divisible by 5 x squared what you do is for the the X variables you choose the smaller of the two so let's take out 5x squared 5x squared divided by 5x squared is 1 negative 15 X cubed divided by 5x squared is negative 3x 5 times negative 3 is negative 15 x squared times X is X cube you can always multiply to check your work if you distribute this is gonna be you should get the original expression so that's how you can tell if your work is correct in this lesson we're going to talk about factoring by grouping typically you'll have a polynomial expression with four terms like this 1x cubed minus 4x squared plus 3x minus 12 now and what it's a factor by grouping you want to separate it into two parts the first two terms take out the GCF the greatest common factor the greatest common factor between X cubed for x squared is x squared X cubed divided by x squared is X negative 4x squared divided by x squared is negative 4 now do the same for the last two take out the GCF the GCF between 3x and negative 12 is stream 3x divided by 3 is X negative 12 divided by 3 is negative 4 now if you get this common term that means you're on a right track at this point you can write 2 parentheses what's going to go in the first one is the X minus 4 what's going to go in the second one is what you see on the outside x squared plus 3 and so that's gonna be the answer now let's try another example 2 R cubed minus 6 R squared plus 5 R minus 15 feel free to pause the video and factor by grouping so let's start with the first two terms the GCF is 2 R squared 2 R cubed divided by 2 R squared is R and negative 6 R squared divided by 2 R squared it's three now for the last two terms the GCF is five so if we take out a five five are divided by five is our negative 15 divided by five is negative 3 by the way how can you know if you can factor by grouping because sometimes you can and sometimes you can if you divide negative six R squared or rather just look at the coefficients if you divide negative six by positive two notice that you'll get negative three and if you divide negative 15 by five you will also get a negative drink when you see that when the first two terms have the same ratio as the last two terms and that means that you can factor by grouping now on the next step let's factor out our -3 if we take out our a minus 3 from the first term we're gonna have to R squared left over and if we take this time divided by the GCF our monastry it's going to give us plus 5 and so this is the solution so now you know how to factor by grouping in this video we're gonna talk about factoring trinomials particularly when the leading coefficient is 1 so here's an example x squared plus 7x plus 12 the leading coefficient is 1 now for these examples all I need to do is find two numbers that multiply to 12 but that adds to the middle term 7 so let's make a list factors of 12 are running 12 12 divided by 2 is 6 12 divided by 3 is 4 1 plus 12 is 13 2 plus 6 is 8 but 3 plus 4 7 so to factor it all it is is just going to be X plus 3 times X plus 4 if you foil this expression you will get the original expression x times X is x squared x times 4 is 4x 3 times X is 3x 3 times 4 is 12 and 4 X plus 3 X is 7x so foiling is just a reverse of factory now let's try another example what about x squared plus 3x minus 20 feel free to pause the video and see if you do this one so what two numbers multiply to negative 28 but adds a 3 well we know that 7 times 4 is 28 so it could be negative 7 and positive 4 or positive 7 and negative 4 negative 7 plus 4 is negative 3 positive 7 plus negative 4 is positive 3 so it's going to be X plus 7 times X minus 4 and that's how you can factor it now let's try this one x squared minus 3x minus 10 so what two numbers multiply to negative 10 but add to negative 3 so this is going to be negative 5 or 2 or 5 and negative 2 now negative 5 plus 2 adds up to negative 3 and it still multiplies to the negative 10 so the answer is X minus 5 times X plus 2 now what about this example of x squared minus 9 X plus 20 what two numbers multiply to 20 but adds a negative 9 so they both have to be negative numbers 20 divided by negative 1 is negative 20 if we divided by negative 2 we're gonna get negative 10 3 doesn't go into 20 but if we divided by negative 4 we'll get a negative 5 and so it's gonna make a list negative 4 plus negative 5 adds up to negative 9 so the answer is X minus 4 times X minus 5 now let's try one more example 2x squared plus 20x plus 48 now you can use the same technique but notice that you can take out the GCF and if you see that you want to do that first so let's factor out a 2 2x squared divided by 2 is x squared xx X divided by 2 is 10 48 divided by 2 is 24 now what two numbers multiply to 24 but adds 10 so if we make a list we have running 24 if we divided by 2 you will get 12 if we divided by 3 it will give us 8 if we divided by 4 will get 6 4 plus 6 is 10 so it's going to be X plus 4 times X plus 6 and with practice you'll be able to do this quickly now what if the leading coefficient is not equal to 1 what should you do well what you want to do is you need to multiply the leading coefficient by the constant term that is you want to multiply 2 times negative 3 2 times negative 3 is negative 6 next find two numbers that multiply to negative 6 but add to negative 5 well we have 2 1 negative 3 and we have 1 and negative 6 but 1 plus negative 6 is negative 5 so what you want to do at this point is replaced negative 5 X with negative 6 X + 1 X because they add up to negative 5 and now that you have four terms factor by grouping in the first two take out the GCF which is 2x - x squared divided by 2 X is X negative 6 X divided by 2 X is negative 3 now in the last two terms if there's nothing to take out and take out on one so if we take out in 1 it's just gonna be the same of X minus 3 now if we factor X minus 3 which we have in both terms we're gonna be left over with 2x plus 1 and so that's how you can factor it and if you want to you can foil eight to check to see if that's the right answer let's try another one 15 x squared plus X minus 6 so let's multiply 15 and negative 6 15 times negative 6 is negative 19 now what two numbers that multiply to negative money add up to positive so we have run 90 is divisible by 2 that will give you 45 if you divide it by 3 you get a negative 34 it is going to 90 if you divided by 5 you will get a negative 18 if you divided by 6 and negative 15 7 and 8 does go into it if you divided by 9 you'll get a negative 10 notice that 9 and negative 10 they differ by one they add up to negative 1 so reverse the sign and they'll add up to positive one so let's replace 1 1 X with 10x and negative 9 X the order in which you place it doesn't matter I just prefer to keep the 15 and the 10 because it's easy to take out a 5 so now let's factor by a grouping in the first two terms let's take out 5 X 15 x squared divided by 5 X is 3 X 10 X divided by 5 X is tuned now in the last two terms let's take out negative 3 negative 9 X divided by negative 3 is 3x negative 6 divided by negative 3 is plus 2 and notice that we have a common term so now let's take out 3 X plus 2 will be left over with 5 X minus 3 and so that is the answer so now you know how to factor a trinomial and when the leading coefficient is not 1 in this lesson we're gonna talk about how to factor a perfect square trinomials a perfect square trinomial is in the form a squared plus 2 a B plus B squared and it's equal to a plus B times a plus B which you can write it as a plus B squared so how can you tell if a perfect square trinomial is a perfect square trinomial so what you want to do is take the square root of this term and the square root of that term multiplied by two you should get the middle term so for example x squared plus 8x plus 16 now for this particular trinomial we can factor it using the lessons that we've learned in 8.3 sometimes however this expression may not be easy to factor and it might be useful to use the equation so let's do it both ways um lesson 8.3 we would find two numbers that multiply to 16 but that add to 8 this would be 4 & 4 4 times 4 is 16 4 plus 4 is 8 so it's going to be X plus 4 times X plus 4 and because they're the same it's X plus 4 squared so using that method is not bad you can quickly get the answer for this type of example but now let's use the equation so first take the square root of 1 the coefficient the square root of 1 is 1 and the square root of 16 is 4 multiply 1 and 4 1 times 4 is 4 and then multiply by 2 if this answer is equal to the middle coefficient then you can factor it's a perfect square trinomial so you can use the equation now that's what I meant to say that was at a loss of words so the equation is a plus the B squared so all you need to do is take the square root of x squared which is X the square root of 16 which is 4 and just write it like this now let's try another example 9x squared plus 6x plus 4 this a perfect square trinomial so the square root of 9 is 3 the square root of 4 is 2 3 times 2 is 6 and you have to double it 6 times 2 is 12 which is not 6 so this is not a perfect square trinomial now what about this one 4x squared plus 12x plus 9 the square root of 4 is 2 the square root of 9 is stream 2 times 3 is 6 and if we double it we do get 12 so that's a perfect square trinomial so the factor in the square root of 4x squared is 2x the square root of 9 is 3 and then it's going to be 2x plus 3 squared now if we were to use the end of the other technique here's what we'll have to do it's going to be a lot more work in this particular example we're gonna have to multiply 4 & 9 4 times 9 is 36 now when you find two numbers that multiply to 36 but add to 12 so we have 2 and 18 we have 4 & 9 there's also 3 times 12 and as you can see it's probably gonna be a fraction so this is a lot much this is hard it's a factor or actually six and six there we go six times six is thirty-six but six plus six as up to 12 so we can replace the middle term with 6x plus 6x now let's take out the GCF then first two terms that's going to be 2x 4x squared divided by 2x is 2x 6x divided by 2x a string for the last two we could take out a 3 6x divided by 3 is 2 X 9 divided by 3 is string so now we can take out 2x + 3 and we're gonna be left with 2x + 3 so as you can see for this example it's easier to use the equation this is a lot more work to factor it this way let's try another example one more at least try this one 9x squared plus 30x plus 25 so first identify if it's a perfect square trinomial and then factor so let's take the square root of 9 just nine that's going to be 3 and the square root of 25 is 5 now 3 times 5 is 15 and 15 times 2 is 30 so it's a perfect square trinomial so now we can use the formula to factor let's take the square root of 9 x squared + 25 the square root of 9x squared is 3x the square root of 25 is 5 and after that 2 squared so that's how you can factor a perfect square trinomial the easy way now what about difference of squares how can we factor those things well if you see something that looks like this a squared minus B squared it's just going to be a plus B times a minus B that's the formula so let's say if we have x squared minus 25 x squared is a perfect square because you can take the square root of it 25 is a perfect square five times five is 25 so what you need to do is take the square root of x squared which is X the square root of 25 which is five and one of them is going to be plus and the other is going to be minus so knowing that try this one x squared minus 36 + 4 x squared minus 9 + 25 x squared minus 81 so feel free to pause the video as you work on these the square root of x squared is X the square root of 36 is 6 and one of them is going to be positive and the other will be negative and that's all you got to do now for the next example the square root of 4x squared is 2x the square root of 9 is 3 so it's going to be 2x + 3 and 2x - 3 the square root of 25 x squared is 5 X the square root of 81 is 9 so the answer is 5 X + 9 + 5 X - 9 now let's work on some more examples now what about these two 3x squared - 48 + 5 x squared - 45 notice that we can't square root 3 or 48 it's not a perfect square it won't give us an integer or a whole number so what we got to do first is take out the GCF which is Street the Reax squared divided by 3 as x squared and negative 48 divided by 3 is negative 16 now we have a difference of perfect squares the square root of x squared is X the square root of 16 is 4 and so it's going to be X plus 4 times X minus 4 over 3 on the outside now for the next one we could take out the GCF swell which is 5 and so we'll be left with x squared - 9 the square root of x squared is X the square root of 9 is 3 so it's going to be X plus 3 times X minus 3 times 5 now what about a bigger example like 16 X to the fourth minus 81 Y to the eighth what can we do here the square root of 16 is 4 the square root of x to the fourth is x squared simply divide the exponent by 2 the square root of 81 is 9 and the square root of Y to the eighth is y to the fourth just divide the exponent by 2 now notice that we could factor this expression it's a difference of perfect squares we can't factor the other one because it's a sum of perfect squares only a difference of squares can we factor the square root of 4x squared is 2x and the square root of 9 Y to the fourth is 3y squared and so this is the answer now in this video we're going to talk about how to factor sums and difference of cubes so you need to know the formulas so let's talk about the sum of cubes a cube plus B cube to factor it it's gonna be a plus B times a squared minus a B plus B squared so while we have the equation on the board let's work on an example so let's say if we have X cubed plus 8 so what that means is that 8 to the third is equal to X cubed if we take the cube root of both sides a is equal to X now B to the third correlates to 8 the cube root of 8 is 2 so and B is 2 so now it's going to be a plus B or X plus 2 a squared that's going to be x times X which is x squared a B that's x times 2 which is 2x B squared that's 2 squared which is 4 and that's how you can factor it using the formula now let's try another example using the difference of squares a cubed minus b cubed so this is gonna be a minus B times a squared plus 2 a B plus B squared let's try two examples let's try 8x cubed minus 27 first so what is the cube root of 8 the cube root of 8 is 2 because 2 times 2 times 2 3 times 8 and the cube root of x cube is X the cube root of 27 is 3 so a is 2x B a stream a squared that's 2x times 2x that's 4x squared oh this is not supposed to be to a B this is supposed to be just a B so just get rid of that now a B is 2x time stream so that's gonna be 6x and make sure to change the sign if you see a negative sign here it's gonna be positive on the right side now B squared that's 3 squared which is 3 times 3 that's 9 so that's the answer let's try this example X cubed minus 1 over 8 the cube root of x cube is X the cube root of 8 is 2 so the cube root of 1 over 8 is 1 over 2 a squared is going to be x squared a times B X times 1/2 that's 1/2 X and then 1/2 squared that's 1 squared is 1 2 squared is 4 so it's 1/4 and that's the answer let's try an example with large numbers 125 X to the 6 minus 64 wide to the knife so the cube root of 125 is 5 the cube root of x to the 6 is x squared divide the exponent by 3 the cube root of 64 is 4 and the cube root of Y to the knife is y cube 9 divided by 3 is 3 now a squared that's going to be 5 x squared times 5 x squared which is 25 X to the 4th and then change the sign plus 5 times 4 is 20 and then x squared times y cube just write them together and then plus B squared B is 4 y cubed 4 times 4 is 16 y cubed times y cube is y to the 6 you got to add the exponents 3 plus 3 is 6 now let's try another one with or using the sum of cubes equation it's always good to write the equation first so you have something to follow along with try this one to 16 X to the twelfth power plus 343 Y to the 15th power the cube root of 216 is six the cube root of x to the 12th is X to the four twelve divided by three is four the cube root of 343 is 7 and Y is the 15 is going to be Y to the fifth power because 15 divided by 3 is 5 a squared that's gonna be 6 times 6 which is 36 X to the fourth times X to the fourth you add a 4 plus 4 and get 8 8 times B 6 times 7 is 42 and then it's going to be X to the fourth Y to the fifth you just multiply these two together anam b squared 7 squared 7 times 7 is 49 and Y to the fifth times y to the fifth that's 5 plus 5 is 10 so this is the answer and now you know how to factor sums and differences of cubes now what if you were to see a problem that looks like this a squared plus 6a plus 9 minus B squared plus 8b minus 16 what would you do in this problem the first thing I would do is take out a negative 1 in the last returns so it's going to be B squared minus 8b plus 16 now notice that we have a perfect square trinomial on the left and on the right the first three terms represent a perfect square trinomial the square root of 1 is 1 the square root of 9 is 3 1 times 3 is 3 times 2 will give us the middle term and on the right side the square root of 16 is 4 times one that's gonna be four times by 2 you're gonna get 8 even though the sign is different it's negative 8 it can still work so now let's go ahead and factor it the square root of a squared is a the square root of 9 is string so it's going to be a plus 3 squared and then minus on the other side the square root of B squared is B the square root of 16 is 4 but because of the negative sign it's not gonna be B plus 4 but it's gonna be B minus 4 squared so now we have a difference of perfect squares so using the formula a squared minus B squared is equal to a plus B times a minus B we could factor it further so a is basically a plus 3 B is like B minus 4 so it's gonna be a plus 3 and then plus B minus 4 and then a plus 3 minus B minus 4 so on the Left let's add so it's gonna be a plus B and we can combine 3 and negative 4 3 plus negative 4 it's negative 1 on the right and we need to subtract it's gonna be a minus B if you distribute the negative sign and then this is plus 3 and then negative negative B is plus 4 so 3 plus 4 is 7 so this is the answer a plus B minus 1 and a minus B plus 7 now let's try another example 4x squared plus 20x plus 25 minus 9 Y squared minus 24 and Y minus 16 so feel free to pause the video and work on this example now the first thing I'm gonna do is I'm gonna take out the negative 1 from the first three terms on the right so it's gonna be positive 9y squared plus 24 y plus 16 now the first we represent a perfect square trinomial the square root of 4 is 2 the square root of 25 is 5 2 times 5 is 10 which is 1/2 at 1 so it's a factor in the square root of 4x squared is 2x the square root of 25 is 5 and just add an exponent of now the next one is a perfect square trinomial the square root of 9 is 3 the square root of 16 is 4 3 times 4 is 12 12 times 2 is 24 the square root of 9 y squared is 3 y the square root of 16 is 4 and so this is what we're gonna have now we have a difference of perfect squares so it's gonna meet 2x plus 5 on both sides and 3y plus 4 so here it's gonna be positive and here it's gonna be negative now on the left let's add 2x plus 3y 5 plus 4 is 9 on the right don't forget to distribute the negative sign so it's gonna be 2x minus 3y and 5 minus 4 is plus 1 and so that's gonna be the answer in this lesson we're gonna talk about how to solve equations by factoring so let's start with a simple example 6x squared minus 30x is equal to 0 find the value of x so here we have a binomial all we can do is take out the GCF the GCF is 6x 6x squared divided by 6x X negative 30x divided by 6x is negative 5 so now we can use the zero product property rule zero times anything is zero and therefore if 6x is equal to zero the whole thing is zero or if X minus five is equal to zero this equation will be true now solving for x we can see that X is equal to zero zero divided by anything is zero and any other equation if we add five we can see that X is equal to five so that's how you can solve an equation by factoring let's try some more examples now what about this one 3 x squared minus 27 what is the value of x notice that we can take out the GCF which is string and we'll be left with squared minus nine so we have a difference of perfect squares so it's a factor in the square root of x squared is X is the square root of nine is three and then we could set each factor equal to zero so X plus three is equal to 0 and next minus three is equal to zero so X is equal to negative three and positive three so that's the solution for this one let's try one more example x squared minus 5x minus 36 so here we have a leading trinomial with a coefficient of one and what two numbers multiply to negative 36 but add to negative five what would you say if we divided by two we'll get negative 18 if we divide it by three we'll get a negative 12 if we divide by four we'll get negative 9/4 and negative 90 differ by five four plus and negative nine is negative five so it's a factor it's gonna be X plus 4 times X minus 9 now let's set each factor equal to zero and now we can find the value of x so X is equal to negative 4 and positive 9 so now you know how to solve equations by factoring