Factoring Trinomials - Part 1
To factor a trinomial in the form of ax squared plus bx plus c we are going to use the "ac" method. Notice that the “a” and the “c” are multiplied together and that's exactly what we are going to do. In this scenario, the "a" equals 2, and the "c" equals negative 12. Multiplied together, they equal negative 24. Now, what we must do is look for factors of negative 24 that add together to be 5. This is what happens in the ac method all the time. So, you want to multiply your "a" and "c" together, and look for factors of “a” times "c" that actually add to be "b". In this scenario, we are going to have a positive 8 and a negative 3 that equal 5. Notice, the 8 times negative 3 still equals negative 24. So the 8 and negative 3 are factors of negative 24 that add together to be 5. So these two guys are our "magic" numbers. So we want to use 8 and negative 3. Let me erase all of this. Let's split up the 5x by using this 8 and this negative 3. So, we have 2x squared plus 8x minus 3x minus 12. Notice, all I did was bring down the 2x squared and the negative twelve, and I just split up the 5x into 8x minus 3x. Now, what I can do is actually do this by grouping considering I have four terms, so I am going to rewrite this with a plus sign in the middle (so it doesn't change the quantity of anything). So, I've got 2x plus 8x plus negative 3x minus 12. Now that I have this plus sign there, I can go ahead and group the first two terms together and the last two terms together. So in this first grouping I can take out a 2x. What's left over is x plus 4. And in this next grouping, I can take out a negative 3, which leaves me with x plus 4. Now notice, these are two terms, and within each term they both have the factor of x plus 4. We're just going to go ahead and factor out the x plus 4. What's left over is this 2x and this negative 3. So x plus 4 times 2x minus 3 is the factored form of 2x squared plus 5x minus 12. Now there is another way to do this. You might have that question of, "What if I changed this 8x and this negative 3x around?" That would be fine. It should turn out exactly the same way. Although when you go through the steps, it's not going to look like these steps. Let's do that so you can see the difference. So first what we are going to put the negative 3 first. We've got the 2x squared minus 3x plus 8x minus 12. Luckily, there is already a plus sign there, so we don't have to worry about rearranging anything. All we have to do is group the first two and the last two together. And so in the first grouping we’re going to go ahead and take out the greatest common factor, which is an x. So we have 2x minus 3 left over. And in this next grouping here, we can actually take out a 4. So when we take out that 4, what we have left over is 2x minus 3. So notice once again, we have a term here and a term here. And, 2x minus 3 is the greatest common factor of those. So what we’re going to do is we’re going to factor that out. 2x minus 3 and then times what’s left over, which is this x and the positive 4. So notice, before, we had the answer x plus 4 times 2x minus 3. And that’s okay; those mean the exact same thing, because you have to remember that property, that “a” times “b” equals “b” times “a.”