in this video we're going to talk about how to factor polynomials so let's start with some simple examples how would you factor this expression 6 x minus 12. all we can do is take out the gcf the greatest common factor which is six now to get what remains divide six x divided by six is x and negative twelve divided by six is negative two so that's all we could do for this example let's try two similar problems let's factor 3x cubed minus 9x squared and also 4x squared minus 12x so with the one on the bottom we need to take out the gcf we could take out a 4 from 4 and negative 12 and we can also take out an x variable because they both contain it 4x squared divided by 4x is x negative 12x divided by 4x is minus 3. so that's how we can factor that expression now for the one above we can also take out the gcf in this case the gcf is going to be 3 x squared we could take out a 3 from 3 and negative 9 and each term has at least two x variables so we can take out an x squared now three x cubed divided by three x squared is x negative nine x squared divided by three x squared is -3 and so that's it for those two examples now the next thing we need to talk about is factor in trinomials this is when we have three terms and we're going to factor it when the leading coefficient is one so what you want to do is find two numbers that multiply to 12 but add to seven two numbers that multiply to 12 but add to seven are 3 and 4. 3 plus 4 is 7 3 times 4 is 12. and so the way you would write your answer is going to be like this it's x plus 3 times x plus 4. and so that's it for that example now here's another one try this one x squared plus 2x minus 15. feel free to pause the video if you want to work on it so what two numbers multiply to negative 15 but add to two we know five times three is negative i mean positive fifteen but we need to add a negative sign if we try negative five and positive three this adds up to negative two but if we try positive five negative 3 it adds up to positive 2 but still multiplies to negative 15. so the answer is going to be x plus 5 times x minus 3. and so that is how we can factor that expression try this one two x squared minus six x minus fifty six now notice that the leading coefficient is not one it's two so what you want to do first is see if there's a gcf that you can factor notice that all of the coefficients are even so we're going to try factoring out a 2. 2x squared divided by 2 is x squared negative 6x divided by 2 is negative 3x negative 56 divided by 2 is negative 28. so we now have a trinomial with a leading coefficient of 1. so now let's find two numbers that multiply to negative 28 but add to the middle coefficient of negative three well we know seven times four is twenty-eight if we try positive seven and negative four that adds up to positive three but negative seven and positive 4 adds up to negative 3. so the answer is going to be 2 times x minus 7 times x plus 4. and so that's it for this example now let's try another simple example let's factor 3x squared minus 18x plus 24. so feel free to pause the video if you want to try it notice that all of the coefficients are divisible by 3. so we're going to take out the gcf which is 3. 3x squared divided by 3 is x squared negative 18x divided by 3 is negative 6x 24 divided by 3 is 8. so now we need to factor this trinomial on the inside of the parentheses two numbers that multiply to 8 but add up to negative 6 are going to be negative 4 and negative two negative four plus negative two adds up to negative six so we can replace x squared minus six x plus eight with x minus four times x minus two and we'll keep the three in the front so this right here is the answer now let's say if you have something like this x squared minus 16 or x squared minus 64. how would you factor these expressions go ahead and try these problems so the situation that we have here is a difference of perfect squares what you want to do first is take the square root of x squared the square root of x squared is x the square root of 16 is 4 and then one of them will be positive and the other will be negative and that's how you can factor an expression like that so for the next one the square root of x squared is x the square root of 64 is eight and we're gonna have x plus eight x minus eight the square root of four is two the square root of x squared is x so the square root of four x squared is two x the square root of 25 is five and so we're going to have 2x plus 5 and 2x minus 5. for the last one the square root of 9x squared is 3x the square root of 49 is 7 and so it's going to be 3x plus 7 times 3x minus 7. so that's how you can factor expressions in the form of a difference of perfect squares now let's say if you have a problem that looks like this 2x squared minus 5x minus 3. what would you do to factor it now notice that there's no gcf that we could take out 2 5 and 3 there's no common number that we can divide to get rid of the 2 in the front so in this situation what we need to do is multiply the leading coefficient by the constant term two times negative three is negative six next find two numbers that multiply to negative six but add to the middle coefficient of negative five this is going to be negative six and plus one negative six times one is still negative six but it adds up to negative five now what we're going to do now is we're going to replace the middle coefficient with negative 6x plus 1x keep in mind the value of the expression is still the same because these two still add up to negative 5x now our next step is something called factoring by grouping in the first two terms you want to take out the gcf which is 2x 2x squared divided by 2x is x negative 6x divided by two x is minus three now on the last two terms take out the gcf there appears to be none which in this case we're going to take out a one so when these two things are the same you're on the right track so now we're going to factor out x minus 3. if we take out x minus 3 from this term we're left with 2x if we take out x minus 3 from that term we're left with plus 1. and so the 2x that you see here goes here and the positive one goes there and so this is the final answer if you want to check the work you can foil this expression but it's going to give you what you started with in the beginning now let's try a similar example for the sake of practice go ahead and factor 6x squared plus x minus 15. so we're going to follow the same procedure we're going to multiply the leading coefficient by the constant term 6 times negative 15 is negative 90. now we need to find two numbers that multiply to negative 90 but add to positive 1. so this is going to be 9 and 10. we're going to use negative 9 and positive 10. negative 9 plus 10 is 1. now our next step is to replace positive 1x with negative 9x and positive 10x and then factor by grouping in the first two terms we can take out a three x six x squared divided by three x is two x negative nine x divided by three x is minus three in the last two terms we could take out a five 10x divided by 5 is 2x negative 15 divided by 5 is negative 3. so now we're going to factor out 2x minus 3. so in the next step you could just write it once and then what you see here is going to go inside of the next parenthesis so that's going to be 3x plus 5. and so that's how you can factor a trinomial where the leading coefficient is not 1 and when you can't take out the gcf in the beginning let's try one more example but this one is going to be different than the previous two so let's say we have 3x cubed minus 2x squared minus 12x plus eight so we have a polynomial expression with four terms how can we factor it the first thing is to notice that the first two coefficients have the same ratio as the last two coefficients that is 3 divided by negative 2 is equal to negative 12 divided by 8. both are equal to negative 1.5 so that means that we could factor by grouping so what we're going to do is take out the gcf in the first two terms which is going to be x squared 3x cubed divided by x squared is 3x negative 2x squared divided by x squared is minus 2. in the last two terms we're going to do the same thing take out the gcf the gcf is minus 4. negative 12x divided by negative 4 is 3x positive 8 divided by negative 4 is negative 2. as we can see these two are the same so we're going to rewrite that in the next line and then what we see here is going to go in the next parentheses so that's going to be x squared minus 4. now notice that we have a difference of perfect squares situation here so we can factor x squared minus 4 like this the square root of x squared is x the square root of 4 is 2 1 will be positive the other will be negative and this is the final answer so that's how you could factor a polynomial expression that looks like that so that's it for this video thanks again for watching for those of you who want more example problems uh check out the links in the description section