Basics of Factoring Trinomials

Welcome to a video that will cover the basics on how to factor trinomials in the form of ax squared plus bx plus c when a is equal to 1, which means the first term will be x squared. We just finished learning how to multiply two binomials together, meaning we were given something in factored form and then we multiplied it out, it was in expanded form. And now we're going to be doing the opposite. We're going to be given a trinomial in expanded form. And we're going to be asked to write it in factored form. So we'll be going from this form to this form. Let's take a look at how we're going to do that. If we have a trinomial when the leading coefficient is 1, meaning it's in the form of x squared plus bx plus c, it's a pretty straightforward process. Here's what we're going to do. We're going to make two sets of parentheses and put the factors of x squared in the first position of each set of parentheses. When the leading coefficient is x squared, there's only two factors of x squared, and that would be x and x. Step two is our main task. In order to find the terms that go in the second positions, we need to find the factors of c that add to b. Because that's so important, let's say that again. We need to find the factors of c that add to b. So for example, if we wanted to factor x squared plus 8x plus 12, We'd start by making two sets of parentheses. The first positions in the binomials come from the factors of x squared. That would be x and x. And now we need to find the factors of positive 12 that add to positive 8. And it is important that we pay attention to the sign. Notice both c and b are positive. So again, we want the factors of positive 12 that add to positive 8. Now if you know your multiplication tables really well, You probably already can tell what those factors would be. If you can't, what you might want to do is just list all of the factors of 12. Meaning it could be 1 times 12, but that doesn't sum to positive 8. 2 times 6, and 2 times 6 does add to positive 8. Therefore, these are the terms that go in the second position. Both of these terms are positive, so we'll have x plus 2 and x plus 6. Now we could also have put the plus 6 here and the plus 2 here. Remember, multiplication is commutative. Now if you don't know your multiplication tables all that well, you might find it helpful to have a multiplication table out. So for example, on the previous problem, if I couldn't find the factors of 12, I could look for 12s in this table. For example, here's a 12, which tells me two factors of 12 would be positive 3 and positive 4. There's another 12 here. That tells me the factors of 12 would be 6 and 2. And the only one that's missing from this table would be 1 times 12, because this table is a 10 by 10 multiplication table. Let's go and take a look at some more examples. Here we have x squared minus 4x minus 32. So since our first term is x squared, we can go ahead and make two sets of parentheses. The first positions come from the factors of x squared. which would be x and x. And now our goal is to find the factors of negative 32 that add to negative four. Again, notice how we do have to pay attention to the sign. So let's go ahead and list some of the factors of negative 32. Could be negative one times 32, or negative 32 times positive one. Either of these factors add to negative four. Let's try negative two times 16, and negative 16 times two. Again, neither of these factors add to positive 4. Let's go ahead and try negative 4 times 8, or maybe negative 8 times 4. And I think we found the winning pair. Negative 8 times positive 4 does give us negative 32. And then when we add these factors, negative 8 plus 4 does give us negative 4. So our two factors are going to be x minus 8. and x plus 4. Again notice if the factor is negative, we're going to write minus. If it's positive, we're going to write plus. Our next example, again the factors of y squared would be y and y. Here we want the factors of positive 36 that add to negative 13. One thing we can notice on this problem is, if we want factors of positive 36 that add to negative 13, the two factors must both be negative to get a negative sum. and a positive product. So we'll have negative one times negative 36. That doesn't add to negative 13. Negative 2 times negative 18, again, doesn't add to negative 13. How about negative 3 times negative 12? Well, that doesn't work either, but I think the next one will. Negative 4 times negative 9 does give us positive 36, and these factors also add to negative 13. So we'll use these for the terms of the second position. So one factor will be y minus 4. The other factor will be y minus 9. Let's go ahead and take a look at a couple more. The factors of x squared will be x and x. And now we want the factors of negative 12 that add to a positive 1. And again, if you know your multiplication tables, you probably can already figure out what that would be. But if not, we're going to go ahead and list them over here on the right. Notice we want the product to be negative. So one factor will be positive. one factor will be negative. So let's see, we have negative one times 12, or negative 12 times one. These do not have a sum of positive one. Negative two times six, or negative six times two. Again, neither of these have a sum of positive one. Next would be negative four times three, or negative three times four. And one of these will work. If we want a sum of positive one, We'll have to have a positive 4 and a negative 3. So our factors will be x minus 3 and x plus 4. Again, we can always check these by multiplying it out. Next, we have x squared plus 28x plus 75. We have x and x. We want the factors of positive 75 that add to a positive 28. In this case, we'll only consider the positive factors of 75, since we know the sum has to be positive. So of course 1 times 75 won't work, but 3 times 25 does give us a positive 75 and also adds to 28. So we'll have x plus 3 times x plus 25. Again, we could put the plus 25 here and the plus 3 here. It won't change the product. Let's take a look at two more. Now notice on these two problems, the leading coefficient is not just x squared or just y squared. Now the reason I put these in here is we can't forget the first step in any factoring problem. And that first step is to look for the greatest common factor. And on this problem here, this 3 is a common factor among all three terms. So we can factor out the 3 first and see what we have left. If we factor out a 3 from x squared, we have x squared. If we factor out a 3 from 27x, that would leave 9x. And if we factor a 3 out of 42, we're left with 14. Notice now that we factored out the greatest common factor, the leading coefficient inside the parentheses is now 1, so we can proceed to try to factor the inner trinomial. Let's go ahead and give it a try. We have a factor of 3 and then two sets of parentheses. The first position would consist of x and x from the factors of x squared. And now we want the factors of positive 14 that add to negative 9. Since the sum has to be negative, we'll only consider when the two factors of positive 14 are negative. So negative 1 times negative 14, that doesn't sum to negative 9. But negative 2 times negative 7 does give us positive 14. And when we sum these, we do get negative 9. So we're going to have x minus 2 and x minus 7. Okay, in our last example, again the leading coefficient is not 1, so we should look for the greatest common factor. Notice all these terms have a factor of 2 as well as a factor of y in common. So we're going to factor out 2y. We'd be left with y squared minus 6y minus 40. And again, notice that the Leading coefficient inside the parentheses is 1, so we can go ahead and factor the inner trinomial like we did the previous examples. So we'll have 2y and then two sets of parentheses. We'll have a y here and a y here. And now we want the factors of negative 40 that add to negative 6. Since the product must be negative, we're going to have to have one positive factor and one negative factor. So let's go ahead and start with negative 2 times 20, or negative 20 times 2. Those don't work because they don't sum to negative 6. Let's go ahead and try negative 4 times 10. or maybe negative 10 times 4. And I think we found it. Negative 10 times 4 does give us negative 40, and the sum is also negative 6. So we'll have y minus 10 and y plus 4. That'll do it for the basics on factoring a trinomial with a leading coefficient of 1. Thank you for watching.

We're going to be given a trinomial in expanded form. And we're going to be asked to write it in factored form. So we'll be going from this form to this form. Let's take a look at how we're going to do that. If we have a trinomial when the leading coefficient is 1, meaning it's in the form of x squared plus bx plus c, it's a pretty straightforward process.

Here's what we're going to do. We're going to make two sets of parentheses and put the factors of x squared in the first position of each set of parentheses. When the leading coefficient is x squared, there's only two factors of x squared, and that would be x and x. Step two is our main task. In order to find the terms that go in the second positions, we need to find the factors of c that add to b.

Because that's so important, let's say that again. We need to find the factors of c that add to b. So for example, if we wanted to factor x squared plus 8x plus 12, We'd start by making two sets of parentheses.

The first positions in the binomials come from the factors of x squared. That would be x and x. And now we need to find the factors of positive 12 that add to positive 8. And it is important that we pay attention to the sign.

Notice both c and b are positive. So again, we want the factors of positive 12 that add to positive 8. Now if you know your multiplication tables really well, You probably already can tell what those factors would be. If you can't, what you might want to do is just list all of the factors of 12. Meaning it could be 1 times 12, but that doesn't sum to positive 8. 2 times 6, and 2 times 6 does add to positive 8. Therefore, these are the terms that go in the second position.

Both of these terms are positive, so we'll have x plus 2 and x plus 6. Now we could also have put the plus 6 here and the plus 2 here. Remember, multiplication is commutative. Now if you don't know your multiplication tables all that well, you might find it helpful to have a multiplication table out.

So for example, on the previous problem, if I couldn't find the factors of 12, I could look for 12s in this table. For example, here's a 12, which tells me two factors of 12 would be positive 3 and positive 4. There's another 12 here. That tells me the factors of 12 would be 6 and 2. And the only one that's missing from this table would be 1 times 12, because this table is a 10 by 10 multiplication table. Let's go and take a look at some more examples.

Here we have x squared minus 4x minus 32. So since our first term is x squared, we can go ahead and make two sets of parentheses. The first positions come from the factors of x squared. which would be x and x. And now our goal is to find the factors of negative 32 that add to negative four. Again, notice how we do have to pay attention to the sign.

So let's go ahead and list some of the factors of negative 32. Could be negative one times 32, or negative 32 times positive one. Either of these factors add to negative four. Let's try negative two times 16, and negative 16 times two. Again, neither of these factors add to positive 4. Let's go ahead and try negative 4 times 8, or maybe negative 8 times 4. And I think we found the winning pair.

Negative 8 times positive 4 does give us negative 32. And then when we add these factors, negative 8 plus 4 does give us negative 4. So our two factors are going to be x minus 8. and x plus 4. Again notice if the factor is negative, we're going to write minus. If it's positive, we're going to write plus. Our next example, again the factors of y squared would be y and y.

Here we want the factors of positive 36 that add to negative 13. One thing we can notice on this problem is, if we want factors of positive 36 that add to negative 13, the two factors must both be negative to get a negative sum. and a positive product. So we'll have negative one times negative 36. That doesn't add to negative 13. Negative 2 times negative 18, again, doesn't add to negative 13. How about negative 3 times negative 12?

Well, that doesn't work either, but I think the next one will. Negative 4 times negative 9 does give us positive 36, and these factors also add to negative 13. So we'll use these for the terms of the second position. So one factor will be y minus 4. The other factor will be y minus 9. Let's go ahead and take a look at a couple more. The factors of x squared will be x and x. And now we want the factors of negative 12 that add to a positive 1. And again, if you know your multiplication tables, you probably can already figure out what that would be.

But if not, we're going to go ahead and list them over here on the right. Notice we want the product to be negative. So one factor will be positive.

one factor will be negative. So let's see, we have negative one times 12, or negative 12 times one. These do not have a sum of positive one. Negative two times six, or negative six times two. Again, neither of these have a sum of positive one.

Next would be negative four times three, or negative three times four. And one of these will work. If we want a sum of positive one, We'll have to have a positive 4 and a negative 3. So our factors will be x minus 3 and x plus 4. Again, we can always check these by multiplying it out. Next, we have x squared plus 28x plus 75. We have x and x. We want the factors of positive 75 that add to a positive 28. In this case, we'll only consider the positive factors of 75, since we know the sum has to be positive.

So of course 1 times 75 won't work, but 3 times 25 does give us a positive 75 and also adds to 28. So we'll have x plus 3 times x plus 25. Again, we could put the plus 25 here and the plus 3 here. It won't change the product. Let's take a look at two more. Now notice on these two problems, the leading coefficient is not just x squared or just y squared.

Now the reason I put these in here is we can't forget the first step in any factoring problem. And that first step is to look for the greatest common factor. And on this problem here, this 3 is a common factor among all three terms. So we can factor out the 3 first and see what we have left.

If we factor out a 3 from x squared, we have x squared. If we factor out a 3 from 27x, that would leave 9x. And if we factor a 3 out of 42, we're left with 14. Notice now that we factored out the greatest common factor, the leading coefficient inside the parentheses is now 1, so we can proceed to try to factor the inner trinomial. Let's go ahead and give it a try.

We have a factor of 3 and then two sets of parentheses. The first position would consist of x and x from the factors of x squared. And now we want the factors of positive 14 that add to negative 9. Since the sum has to be negative, we'll only consider when the two factors of positive 14 are negative. So negative 1 times negative 14, that doesn't sum to negative 9. But negative 2 times negative 7 does give us positive 14. And when we sum these, we do get negative 9. So we're going to have x minus 2 and x minus 7. Okay, in our last example, again the leading coefficient is not 1, so we should look for the greatest common factor.

Notice all these terms have a factor of 2 as well as a factor of y in common. So we're going to factor out 2y. We'd be left with y squared minus 6y minus 40. And again, notice that the Leading coefficient inside the parentheses is 1, so we can go ahead and factor the inner trinomial like we did the previous examples. So we'll have 2y and then two sets of parentheses.

We'll have a y here and a y here. And now we want the factors of negative 40 that add to negative 6. Since the product must be negative, we're going to have to have one positive factor and one negative factor. So let's go ahead and start with negative 2 times 20, or negative 20 times 2. Those don't work because they don't sum to negative 6. Let's go ahead and try negative 4 times 10. or maybe negative 10 times 4. And I think we found it. Negative 10 times 4 does give us negative 40, and the sum is also negative 6. So we'll have y minus 10 and y plus 4. That'll do it for the basics on factoring a trinomial with a leading coefficient of 1. Thank you for watching.

Transcript for:Basics of Factoring Trinomials

Transcript for:
Basics of Factoring Trinomials