 So let's try to factor x squared minus x minus 12 and so remember if I want to factor a monic Polynomial I want to look for two numbers that multiply to the constant term So we want to find numbers that multiply to negative 12 and these numbers are So remember the only way to determine if a factorization works is to check it So we try one and negative 12 first. So Is it true that x squared minus x minus 12 is the same as x plus 1 times x plus negative 12 Now before we check our factorization, let's clean up our factors a little bit. Remember that a plus negative b is the same as a Minus b and so instead of writing x plus negative 12. I can write this second factor as x minus 12 And so let's see if this factorization works by expanding And that's not what we want So when a negative 12 don't work So remember factoring requires patience and persistence There is no substitute and so even though one and negative 12 didn't work We can try something else about two and negative six And so the question is is x squared minus x minus 12 the same as x plus 2 times x minus 6 and We'll expand and it isn't We'll try 3 and negative 4 next is x squared minus x minus 12 the same as x plus 3 times x minus 4 so we'll expand and It is and so there's our factorization Now there is one shortcut to factoring notice that the last thing we try always works So you should always start with the last thing Okay, maybe you can't actually start with the last thing because you don't know what that last thing is going to be And so we might ask the following question if we're factoring a monic polynomial We can go through every pair a b that multiplies to q our constant term and see if our polynomial Factors as x plus a times x plus b But what if a lot of numbers multiply to q is there an easier way than checking every possibility? No But we might be able to make better choices For example, let's try to factor x squared minus 14x plus 48 So we want to find two numbers a and b where the product is 48 and to our possibilities are and Don't forget. We also have the possibility of negative numbers that multiply it to 48 So we get a second set of possibilities as well That's a lot of things to go through and we have to check every one. So we'll try 1 and 48 and see if x squared minus 14x plus 48 factors as x plus 1 times x plus 48 So we expand and find Now first of all this didn't work And so we could just move on to trying the next pair of factors But let's do a little bit of analysis and see if we can make better choices. Let's see why this didn't work So the first thing to notice here is that the x term is added plus 49x Because both factors plus 1 plus 48 are positive But we wanted the x term subtracted. We wanted to get minus 14 x And if we try any of the other positive factors 2 and 24 3 and 16 and so on We're still going to get a plus some number of x and so we'll focus on the negative factors We'll ignore all of these positive factorizations The other thing you might notice is the x term itself 49x is large because one of the factors 48 is also a large number But we don't want so large of an x term. We want a relatively small x term So let's try the smallest pair of factors negative 6 and negative 8 So we try negative 6 and negative 8 and we find We get a factorization Or let's take a look at x squared plus 8x minus 48 So again, we list the numbers that multiply to negative 48 and We'll try 1 and negative 48 which fails But again, we might analyze our result a little bit to see why it failed And the thing to notice here is the x term is subtracted minus 47x and that's because the negative factor minus 48 is Larger than the positive factor 1 But we wanted an added x term. We wanted to get plus 8x and So that means we should start looking among the factors where the negative factor is smaller than the positive factor And we can also ignore the factors where the negative factor will be larger And we can also check on the magnitude So notice the magnitude of the x coefficient is large negative 47 because the two factors minus 1 and 48 were very different But we want a smaller x coefficient. So we'll check the factors that are closer together So how about negative 6 and 8 and since this is the second thing we try we know that this is the factorization Well, let's actually check it to see if it works And it doesn't Well, we have other possibilities. We check negative 4 and 12 And finally we get something that works. So again remember Factoring requires patience and persistence. There is no substitute