Fractions involving Quadratic and Linear Terms: Common Factors

Fractions involving Quadratic and Linear Terms Common Factors
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All right! Again, we’ve got a fraction and you can see we have a quadratic on the numerator. So again our cross method. So my two x’s, it’s positive 20, so it’s positive, so it could be positive positive or negative negative but since this is negative we must have negative negative. So I’m gonna start with my factors of negative 4. So the other one will be 5, negative 5. So I’ll start with this one first. x times negative 4 which is negative 4x and as I said the other one will be negative 5 and draw my cross will be negative 5x and you can see that negative 4x plus minus 5x is negative 9x which is what we’ve got there, so we’ve got the right fractions we’ve got the right sorry factors. So the numerator will now become x minus 4, x minus 5, put that on the numerator and see what I did to the denominator, see how 4 is a common factor. So I simply factorized by 4, took the 4 out, and factorized. Because 4 is the common factor. So make sure you don’t forget to factorize the denominator.

Even though it’s not quadratic. It’s a linear expression. So make sure you always take the common factor out. Now you can see that x minus 4, x minus 4. I’ve got common! So all you need to do is cross them out, that’s my favorite part. And now the answer is simply x minus 5 over 4. That’s the answer. Okay? So you just need to leave whatever’s remaining. All right! What I’m gonna do is have a look! A lot of you will start by using the cross method straight away but smart students, so if you are looking at that in a smarter way you can see that I can first factorize by two because two is the common factor, okay? So make sure you do that otherwise, it gets a lot more confusing. So take the two out because it’s common and you can see that four is also common on the denominator, so I can take the four out as well on the denominator, okay?

So trying to take out the common factor all the time first, okay? And then we can worry about the quadratic inside the brackets. So I’m going to use my x and x to use the cross method for the inside part and as I said it’s a positive 20, so it could be positive positive negative negative but because this is negative, it’s going to be negative negative, isn’t it? So I use negative 4 because that’s 4 as a factor of 20 and then x times negative 4 is negative 4x, so the other one will be negative 5. Cross, negative 5x and you can see that negative 4x minus 5x is negative 9x which is what we want. So we’ve got the right form. So the numerator now becomes x minus 4, x minus 5, and I got rid of these because I can cancel that’s going to be 1 and that’s going to be 2, isn’t it? Because 2 and 4 have a common factor of 2, so we don’t it’s just a 1. So we don’t have to put anything there. So basically it’s just this part there which we change to x minus 4, x minus 5, okay? And then we have the 2 x minus 5 left on the denominator.

And can we cross anything out guys? Of course, we can these two are common, so it’s just x minus 4 and 2 left, okay? So it’s actually pretty fun if you can factorize it um correctly. All right! Simplify, so now every single one of you should be able to do this without going straight to the cross method. What can we factorize by first? Are there any common factors? Yes! 3 but what I’m going to do I’m not just going to take out 3. I’m going to take out negative 3. I don’t know if anyone remembers but I always try to stress out that I want the x square to have a positive coefficient. So I don’t want negative x squared, I always want positive x squared. That’s why I take the whole thing out, whole negative 3, okay? So this becomes positive and positive, okay? Because we switched the signs and that’s 6 and 8 because we took out 3, okay?

That’s the common factor and of course the denominator we can take out 2 as well which is the common factor. So let’s go ahead and factorize that inside part, inside quadratic. So draw my x’s, it’s all positive, so my factors must be all positive, so I use 2, x times 2 will be 2x and the other will be 4, cross, 4x, and you can see that 2x plus 4x is 6x which is what we want, so we’ve got the right thing. So I can say it’s negative 3, so leave that thing out there, and then we’ve got x plus 2, x plus 4, that’s how we factorize that big bracket and you can see that I can cancel x plus 4 because they’re common, so we just have that left, okay? Make sure you just keep it, keep those constants outside you don’t have to expand it because our the factorized the form is always the simplest form guys, so don’t go ahead and expand that for some reason, okay? Always leave it in factorized form


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