Factor the polynomial. 3x2−2x−53x^2-2x-53x2−2x−5

( x + 1 ) ( 3 x − 5 ) \left(x+1\right)\left(3x-5\right) ( x + 1 ) ( 3 x − 5 )

Factor the polynomial. 3 x 2 − 2 x − 5 3x^2-2x-5 3 x 2 − 2 x − 5

Alright everyone, so let's take a look at this, this practice problem here. We're gonna factor this polynomial 3x2-2x-5, and what you're gonna see if you try to use the greatest common factor in some of the other methods and like that is that none of them will work. So we're gonna have to use the AC method for this. So I've got my a and my c term over here, and this is my b term. So let's just go ahead and stick to the steps, right? We're gonna have to build our table of all the factors that lead to 'ac'. So 'ac' in this case, I have 3 times 5 sorry, 3 times negative 5, and that's going to give me negative 15. So, I have to look for numbers that multiply to negative 15. Alright, so let's do that. So, let's just do 1. What do I have to multiply 1 by to get to -fifteen? -fifteen, but then I have to flip the sign. So in other words, 1 times -fifteen works, but -one times positive fifteen will also work. Alright? So what about 2? 2 doesn't work, what about 3? Well, we'll see that 35, but I have to do 3 times negative 5. That will work. What about negative 35? I have to flip one of the signs. That will work as well. And actually, that is only those are the only unique possibilities because then I'll just do 5 and negative 3, and we've already covered that pair before. So these are our pairs. So I have to look for which one of these pairs adds to the b term. In my case, the b term is just negative 2. So b is negative 2 over over here. So one and -fourteen gives me negative 4 I'm sorry negative one and -fifteen it gives me negative 14. Negative one and positive 15 give me positive 14. So none of those things will work. What about 3 and -five that will give me -two? So that looks like it works. That's what we're looking for over here. So these are our answers over here because negative 3 and positive 5 gives us 2, that doesn't work. So this is my P and this is my Q, and so what happens is I have to split this out into 4 turns. So the a drops down to 3x2 minuteus actually, I'm gonna do plus 3x minus 5x, and then I do minus 5 over here. Alright? So here what happens is I've just split the, the negative two x into the two terms over here, the two middle terms. And now we can just go ahead and group these this expression. So we can group these things into 2 terms over here, and let's figure out what's common between the first two. So this one should be, you know, you should see that I have a 3x squared in this term, and then I have a 3x in this term. So in other words, what's common between them is that I I actually have a a factor of 3x that's common in both of them. So in other words, that thing can just be brought out to the outside of the parentheses. What am I left with? I'm left with 3x, and then I have basically just x+3x. And remember, when I completely divided something out of a term, there's kind of like a one that's left over over here. Distribute this back and you should get 3x2+3x, because this 3x3xdistributes into 1. Alright. So what about this expression over here, negative 5x and then minus 5? Well, one of the things that you should see here is that really this is just negative 5 times x plus, and then this is negative 5. So actually what happens is that the negative 5 is common in both of them. They both have the same number and the same sign. So that can be brought to the outside as well. And what happens? I basically have '+' and this just becomes negative 5 over here. And then what's left over? Well, so I really could just sort of simplify this to minus 5. And then what happens is I just have 1x over here and the same thing happened. If I divide this negative 5 out, what am I left with over here? Nothing. So I just put a +1. And so that's how to simplify that expression. Again, distribute this back and you should get back to the original expression over here. So now what I can see is that I've actually ended up with the same terms in both of the groups and now I can pull that out to a greatest common factor as well. So I pull this out to the outside of the parentheses and this becomes x+1 and then 3x-five. If you foil this out, you'll get back to your original expression. So this is our final answer. Alright. So that's how to factor these kinds of polynomials where a is not equal to 1. Follow this sort of step by step approach, and you'll always get the right answer instead of having to resort to kind of like a guess and check method. Thanks for watching.