Simplifying rational expressions
When we first started learning about fractions or rational numbers, we learned about the idea of putting things in lowest terms. So if we saw something like 3, 6, we knew that 3 and 6 share a common factor. We know that the numerator, well, 3 is just 3, but that 6 could be written as 2 times 3. And since they share a common factor, the 3 in this case, we could divide the numerator by 3 and the denominator by 3, or we could say that this is just 3/3, and they would cancel out. And in lowest terms, this fraction would be 1/2. Or just to kind of hit the point home, if we had 8/24, once again, we know that this is the same thing as 8 over 3 times 8, or this is the same thing as 1 over 3 times 8 over 8. The 8's cancel out and we get this in lowest terms as 1/3. The same exact idea applies to rational expressions. These are rational numbers. Rational expressions are essentially the same thing, but instead of the numerator being an actual number and the denominator be an actual number, they're expressions involving variables. So let me show you what I'm talking about. Let's say that I had 9x plus 3 over 12x plus 4. Now, this numerator up here, we can factor it. We can factor out a 3. This is equal to 3 times 3x plus 1. That's what our numerator is equal to. And our denominator, we can factor out a 4. This is the same thing as 4 times 3x. 12 divided by 4 is 3. 12x over 4 is 3x. Plus 4 divided by 4 is 1. So here, just like there, the numerator and the denominator have a common factor. In this case, it's 3x plus 1. In this case, it's a variable expression. It's not an actual number, but we can do the exact same thing. They cancel out. So if we were to write this rational expression in lowest terms, we could say that this is equal to 3/4. Let's do another one. Let's say that we had x squared-- let me see a good one. So let's say we had x squared minus 9 over 5x plus 15. So what is this going to be equal to? So the numerator we can factor. It's a difference of squares. We have x plus 3 times x minus 3. And in the denominator we can factor a 5 out. This is 5 times x plus 3. So once again, a common factor in the numerator and in the denonminator, we can cancel them out. But we touched on this a couple of videos ago. We have to be very careful. We can cancel them out. We can say that this is going to be equal to x minus 3 over 5, but we have to exclude the values of x that would have made this denominator equal to 0, that would have made the entire expression undefined. So we could write this as being equal to x minus 3 over 5, but x cannot be equal to negative 3. Negative 3 would make this zero or would make this whole thing zero. So this and this whole thing are equivalent. This is not equivalent to this right here, because this is defined that x is equal to negative 3, while this isn't defined that x is equal to negative 3. So to make them the same, I also have to add the extra condition that x cannot equal negative 3. So likewise, over here, if this was a function, let's say we wrote y is equal to 9x plus 3 over 12x plus 4 and we wanted to graph it, when we simplify it, the temptation is oh, well, we factored out a 3x plus 1 in the numerator and the denonminator. They cancel out. The temptation is to say, well, this is the same graph as y is equal to the constant 3/4, which is just a horizontal line at y is equal to 3/4. But we have to add one condition. We have to eliminate-- we have to exclude the x-values that would have made this thing right here equal to zero, and that would have been zero if x is equal to negative 1/3. If x is equal to negative 1/3, this or this denominator would be equal to zero. So even over here, we'd have to say x cannot be equal to negative 1/3. That condition is what really makes that equal to that, that x cannot be equal to negative 1/3. Let's do a couple more of these. And I'll do these in pink. Let's say that I had x squared plus 6x plus 8 over x squared plus 4x. Or actually, even better, let me do this a little bit. x squared plus 6x plus 5 over x squared minus x minus 2. So once again, we want to factor the numerator and the denonminator, just like we did with traditional numbers when we first learned about fractions and lowest terms. So if we factor the numerator, what two numbers when I multiply them equal 5 and I add them equal 6? Well, the numbers that pop in my head are 5 and 1. So the numerator is x plus 5 times x minus 1. And then our denonminator, two numbers. Multiply negative 2, add a negative 1. Negative 2 and positive 1 pop out of my head. So this is a positive 1, right? x plus 5 times x plus 1, right? 1 times 5 is 5. 5x plus 1x is 6x. So here we have a positive 1 and a negative 2. So x minus 2 times x plus 1. So we have a common factor in the numerator and the denonminator. These cancel out. So you could say that this is equal to x plus 5 over x minus 2. But for them to really be equal, we have to add the condition. We have to add the condition that x cannot be equal to negative 1 because if x is equal to negative 1, this is undefined. We have to add that condition because this by itself is defined at x is equal to negative 1. You could put negative 1 here and you're going to get a number. But this is not defined at x is equal to negative 1, so we have to add this condition for this to truly be equal to that. Let's do a harder one here. Let's say we have 3x squared plus 3x minus 18, all of that over 2x squared plus 5x minus 3. So it's always a little bit more painful to factor things that have a non-one coefficient out here, but we've learned how to do that. We can do it by grouping, and this is a good practice for our grouping, so let's do it. So remember, let's factor 3x squared plus 3x minus 18. So you need to think of two numbers. This is just a review of grouping. You need to think of two numbers that when we multiply them are equal to 3 times negative 18, or it's equal to negative 54, right? That's 3 times negative 18. And when we add them, a plus b, needs to be equal to 3x because we're going to split up the 3x into an ax and a bx. Or even better, not 3x, equal to 3. So what two numbers could there be? Let's see, our times tables. Let's see, they are three apart. One's going to have to be positive and one's negative. 9 times 6 is 54. If we make the 9 positive and we make the b negative 6, it works. 9 minus 6 is 3. 9 times negative 6 is negative 54. So we can rewrite this up here. We can rewrite this as 3x squared, and I'm going to say plus 9x minus 6x minus 18. Notice, all I did here is I split this 3x into a 9x minus 6x. The only difference between this expression and this expression is that I split the 3x into a 9x minus 6x. You add these two together, you get 3x. The way I wrote it right here you can actually ignore the parentheses. And the whole reason why I did that is so I can now group it. And normally, I decide which term goes with which based on what's positive or negative or which has common factors. They both have a common factor with 3. Actually, it probably wouldn't matter in this situation, but I like the 9 on this side because they're both positive. So let's factor out a 3x out of this expression on the left. If we factor out a 3x out of this expression this becomes 3x times x plus 3. And then on this expression, if we factor out a negative 6, we get negative 6 times x plus 3. And now this is very clear our grouping was successful. This is the same thing as-- we can kind of undistribute this as 3x minus 6 times x plus 3. If we were to multiply this times each of these terms, you get that right there. So the top term, we can rewrite it as 3x minus 6-- let me do it in the same color. So we can rewrite it as 3x minus 6 times x plus 3. That's this term right here. I don't want to make it look like a negative sign. That's that term right there. Now let's factor this bottom part over here. Scroll to the left a little bit. So if I want to factor 2x squared plus 5x plus 3, I need to think of two numbers that when I take their product, I get 2 times 3, which is equal to 6, and they need to add up to be 5. And the two obvious numbers here are 2 and 3. I can rewrite this up here as 2x squared plus 2x plus 3x plus 3, just like that. And then if I put parentheses over here, and I decided to group the 2 with the 2 because they have a common factor of 2, and I grouped the 3 with the 3 because they have a common factor of 3. This right here is 2 and a 3. So here we can factor out a 2x. If you factor out a 2x, you get 2x times x plus 1 plus-- you factor out a 3 here-- plus 3 times x plus 1. And our grouping was successful. This is clearly-- let me switch colors-- this is the same thing as 2x plus 3 times x plus 1. So here we've been able to factor it as well. We were able to factor out the denominator as well. Actually, I just realized that I made a mistake. I wrote here minus 3. I wrote a plus 3 over here. Let me backtrack this. That would have been a horrible mistake. I would have had to redo the video. Let me clear all of this, all of this business over here. Let me clear that. This is 2x squared plus 5x minus 3. So once again, a times b needs to be equal to negative 3 times 2, which is negative 6. And a plus b needs to be equal to 5. So in this situation, it looks like if we went with 6 and negative 1, that seems to be a better situation. 6 minus 1 is 5. 6 times negative 1 is negative 6. So that would have been a horrible mistake. So we can rewrite this up here as 2x squared, and I'll group the 6 with the 2x squared because they share a common factor. So plus 6x minus x, this is the same thing as 5x minus 3. I just had to find the numbers to split this 5x into. But 6x minus x is 5x. And if I put some parentheses here, I can factor out of 2x out of this first term. I get 2x times x plus 3. And here I can factor out a negative 1, so minus 1 times x plus 3. And then our grouping was successful. We get 2-- let me do this in a different color-- we get 2x minus 1 times x plus 3. So our denominator here is equal to 2x minus 1 times x plus 3. And once again, we have a common factor in our numerator and our denonminator, the x plus 3. But we have to add the condition that x cannot be equal to negative 3, because that would make this whole thing equal to zero. Or not equal to zero, it would make us divide by zero, which is undefined. So we have to say that x cannot be equal to negative 3. So this expression up here is the same thing as 3x minus 6 over 2x minus 1, granted that we also imposed the condition that x does not equal negative 3. Hopefully, you found that interesting.