Fractions involving Quadratic Terms: Monic Expressions

Fractions involving Quadratic Terms Monic Expressions
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The numerator, first of all, I’m going to use x and x because it’s negative. I know I’m going to use positive and a negative number. I’m going to start with a positive 4. So x times positive 4 is 4x. The other one will be negative 1. Now as I said guys some of you might use negative 4 and positive 1 but as you can see, we need 3. We need a positive 3. So that’s why I know that it’s going to be positive 4 and negative 1 because 4 minus 1 is 3, okay? So try to think about it before you just put in the numbers okay?

So x times negative 1 is negative x. So 4x minus x is 3x which is what we want. So we’ve got the right one for the numerator. Now for the denominator, put x and x because it’s positive 2, it could be positive positive or negative negative because that’s negative we know that’s going to be negative negative. So I’m going to use negative 2 which is going to be negative 2x and then the other one will definitely be negative 1. So that will be negative x. And you can see that negative 2x minus x is negative 3x which is what we want.

So you can see that on the numerator, it’s going to be x plus 4, x minus 1, and on the denominator, it will be x minus 2, x minus 1 just like that, okay? So we can cancel out the common ones, so we’ve just got those leftover that is the answer.

Question 13! Okay! Let’s try again, so x and x for the numerator, 24, it’s negative, so I’m going to use positive and negative. So I’m going to start by using positive 3 and because that’s 3x and negative 8 which is negative 8x. And check! 3x minus 8x is negative 5x which is what we want, so we’ve got the right one for the numerator. Now for the denominator, it’s all positive. So I know all the signs are going to be positive. So x and x and I’m going to use 2 which makes 2x and 3 which makes 3x and add them together you get 5x which is exactly what we want. So I’ve got the right one for both numerator and denominator.

So x plus 3, x minus eight on top, and x plus two, x plus three on the denominator and can we cross anything out? Yes we can, these ones, so we’ve just got that leftover, okay? How repetitive is this? Guys from now on, don’t just wait for me to finish the solutions. Try and pause for a bit and try to do your own factorization so you can practice and check your answers, okay? So please give that a shot the next few questions are shot on your own, okay?

So okay, we’ve got something a little bit different here. What’s happening? We’ve got some extra pronumerals. We’ve got another pronumeral a. But just treat it like you normally do guys don’t worry about the new kind of form. Just treat it the same way. So I’m going to start by expanding it out first of all, just so I can gather some like terms. So I just expanded this that’s all I did. 2 times x and that becomes negative ax and the same for the denominator, just expanded it.

The reason why I did that is now I can gather these two together here and these two together
here so all I did is factorize individually. Remember how when we have four terms or four terms in an expression, I can gather twos together and twos together and factorize individually, that’s what I did. So see how here I can factorize for x which is the common factor and here I factorize by negative a, so that becomes a positive, okay? Make sure you change that to a positive. And same for the denominator, factorize this part by x and factorize this part by a, okay? So gather twos together I group them into twos.

Now! Have a look! You can see that here on the numerator, x plus two is common, and the same for the denominator, x plus 2 is common. So I can take that out as a common factor like that, so here x plus 2 is common, so x plus 2 and we’ve got… I’m going to rub this out for a second…. I take out the x plus 2 because that’s common and therefore we just have x minus a left inside the brackets and the same for here from here and here, x plus 2 is common. So I take that out and we’ve just got x plus a left inside the brackets. And now you can see that x plus 2 is common, so we can cancel them which is good. So we’ve just got x minus a, x plus a, that is our answer for this part. So you don’t really need to use any quadratic formulas but sorry quadratic factorizations. But remember how we did these kinds of ones before when we’re factorizing ones with four terms, so you just pretty much treat the same way as we did in the ones with the four terms, okay? So trying to group them into two.


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