When we first started talking about fractions and rational numbers, we learned about the idea that things they must be of the lowest possible value. If we see something like 3/6, we know that 3 and 6 have a common factor. We know the numerator – 3 is 3, but 6 can be saved as 2 times 3. And since they have a common multiplier, in this case 3, we can divide the numerator by 3 and the denominator by 3, or let’s say it’s just 3/3 and they will be shortened. So the lowest members here are 1/2. And just to make it even clearer, if we have 8/24, we know that this is the same as 8 over 3 8 or the same as 1 over 3 8 on 8 The 8s are shortened and we get 1/3. The same idea can be applied to rational expressions. These are rational numbers. Rational expressions are basically the same thing. But instead of the numerator and denominator real numbers, they are expressions that include variables. Let me show you what I’m talking about. Let’s say I have 9x plus 3 over 12x plus 4 We can decompose this numerator into factors. We have a multiplier of 3. This is equal to 3 by 3x plus 1. Here is what our numerator is equal to. And in the denominator we find a multiplier of 4. This is the same as 4 by 3x. 12 divided by 4 is 3. 12x divided by 4x is 3x. Plus 4 divided by 4 is 1 So here, as in the example, the numerator and the denominator have a common multiplier. In this case, that’s 3x plus 1. In this case we have an expression with a variable, not a number, but we do the same. They are shortened. If we want to write this rational expression with the lowest values, we can say that it is equal to 3/4. Let’s take another example Let’s see a suitable example Let’s say we have x squared minus 9 over 5x plus 15 What will this be equal to? – We are looking for multipliers in the numerator. This is the difference of squares. We have x plus 3 over x minus 3. And in the denominator we can export 5. That’s 5 over x plus 3. Again we have a common multiplier in the numerator and in denominator, so they are abbreviated. We talked about this a few videos ago – we have to be very careful. We can shorten them. We can say that this is going to be equal to x minus 3 over 5, but we must exclude the values \u200b\u200bof x, which will make the denominator 0 and thus will do the whole expression indefinite. We can write this as equal to x minus 3 on 5, but x cannot be equal to minus 3 Minus 3 would make this 0 So this and all this are the same. It’s not the same as here, because here it is determined that x is equal to -3, but it is not here determined that x is equal to – 3. To make them the same, I need to add and the condition that x cannot be -3. Similarly here, if it was a feature, yes let’s say we have y is equal to 9x plus 3 over 12x plus 4 and we want to draw this. When we simplify, it seduces us the idea to export a multiplier of 3x plus one in the numerator and denominator. They are shortening. We want to say, well, it’s the same graphics, where y is equal to the constant 3/4, which is just a horizontal line in y equal to 3/4 But we must add one condition We need to exclude the values \u200b\u200bof x that would do this thing 0, and this would be 0 if x is equal to minus 1/3 If x is equal to minus 1/3, this or that denominator will be equal to 0 So even here we will have to say that x cannot be equal to minus 1/3. This condition allows this to be really equal to but x cannot be equal to minus 1/3. Let’s make a few more examples. I’ll take pink. Let’s say I have x squared plus 6x plus 8 over x squared plus 4x. Or, better yet, let me do it x squared plus 6x plus 5 over x squared minus x minus 2 Again, we want to decompose the numerator and the denominator, point as we did with traditional numbers, when we first learned about fractions and lowest values. Let’s see the numerator – which two numbers, when I multiply them, are equal to 5 and when I add them up are equal to 6? The numbers that come to mind are 5 and 1. So the numerator is x plus 5 over x minus 1. And then, in the denominator we have two numbers that are multiplied give minus 2 and collected give minus 2. I remember -2 and plus 1. So that’s our plus 1, right? x plus 5 by x plus 1, right? 1 in 5 is 5 5x plus 1x is 6x So here we have plus 1 and minus 2, x minus 2 over x plus 1. We have a common multiplier in the numerator and denominator. They are shortened. We can say that this is equal to x plus 5 on x minus 2 But to be truly equal, you have to to add the condition that x cannot be equal to minus 1, because if x is equal to minus 1, this is indeterminate for us. We have to add this condition because this in itself is determined for x is equal to minus 1. We can put minus 1 here and we will get some number. But this is not determined for x equal to minus 1, then we need to add this condition to be able to that really equals that Let’s try something more complicated. Let’s say we have 3x squared plus 3x minus 18, everything on 2x squared plus 5x minus 3. Multipliers are always harder to find things that have a coefficient other than 1, but we have learned how to deal with it. We can do this by grouping and this is a very good exercise, so, let’s do it. Remember, we want to factorize 3x plus 3x minus 18. We need to find two numbers. This is just a group negotiation. We have to think of 2 numbers that are multiplied equal to 3 minus 18, or equal at minus 54, right? That’s 3 minus 18 And when we add them up, a plus b has to be equal to 3x, because we will divide 3x by ax and bx Or better yet, not 3x, equal to 3 What could these numbers be? Let’s see our multiplication tables One number will have to be positive, and the other – negative. 9 over 6 is 54. If 9 is positive for us and we do b is -6, that works. 9 minus 6 is 3. 9 times -6 is -54. We can copy this up here. We can rewrite it as 3x squared, and I will say plus 9x minus 6x minus 18 Notice that here you just presented 3x as 9x minus 6x. The only difference between these two expressions is that presented 3x as 9x minus 6x. We add these two and we get 3x. In fact, as I wrote it down, we don’t need the brackets. And the only reason I do that is so I can group now. And I usually decide how to group the members, looking at what is positive and what is negative, or where there are common factors. Both have a common factor of 3. In fact, it may not matter in this situation, but I want 9 to be on this side because both are positive. Subtract 3x as a multiplier from this expression on the left. When we express this 3x, the expression becomes 3x by x plus 3. And then in this expression, if we put -6, we will get minus 6 over x plus 3. It is clear that our grouping was successful. This is the same as unopened brackets. 3x minus 6 over x plus 3. If we had to multiply this by all these terms, we would reach that. We can rewrite the above term as 3x minus 6 – let me do it in the same color. 3x minus 6 over x plus 3. It is this article here. I don’t want this to look like a minus. It is this article here. Let’s find the multipliers in this part below. I move a little to the left. If I want to factorize 2x plus 5x plus 3, I have to to think of two numbers that multiplied give 2 by 3, which is 6, and their sum is 5. The two obvious numbers here are 2 and 3. I can rewrite this above as 2x squared plus 2x plus 3x plus 3, that’s it And then, if I put parentheses here and decide to group 2 with 2 because they have a common factor of 2, and yes I group 3 with 3 because they have a common multiplier 3. This here is 2 and 3. So here we can export a multiplier of 2x. If we export 2x, we get 2x by x plus 1 plus – here we subtract 3 – plus 3 by x plus 1. And our grouping is successful. It is clear that this (let me change the color) is the same thing as 2x plus 3 over x plus 1. Here, too, we decomposed. We also decomposed the denominator. In fact, I now see that I made a mistake. I wrote minus 3 here, and here – plus 3 Let me go back. That would be a terrible mistake. I would have to shoot the video again. Let me delete all this here. That’s 2x squared plus 5x minus 3 Once again, a b must be equal to minus 3 2, which is minus 6. And a plus b must be equal to 5. In this situation, it seems to be best let’s take 6 and minus 1. 6 minus 1 is 5. 6 minus 1 is minus 6. So that would be a huge mistake … We can rewrite this above as 2x squared and it will we group 6 by 2x squared, because they have a common multiplier So, plus 6x minus x is the same as 5x minus 3. I just need to divide this 5x But 6x minus x is 5x. And if I put brackets here, I can take out 2x from the first article. I get 2x by x plus 3 And here I can subtract minus 1, then minus 1 by x plus 3 And our grouping is successful. To take another color – we get 2x minus 1 by x plus 3. So our denominator here is 2x minus 1 by x plus 3. And again we have a common multiplier in the numerator and the denominator, x plus 3. But we must add the condition that x does not can be minus 3, because then the whole this thing would be 0. Or it would make us divide by 0 and to obtain an indefinite number. So we have to say that x cannot be equal to -3. So the expression here above is the same as 3x minus 6 on 2x minus 1, as long as it is fulfilled the condition that x is not equal to –3. I hope you found this interesting.