Video: Simplifying a Quotient of Two Rational Functions

Simplify the function 𝑛(𝑥) = (𝑥² + 7𝑥)/(6𝑥² + 25𝑥 + 4) ÷ (6𝑥² − 𝑥)/(36𝑥² − 1).


Video Transcript

Simplify the function 𝑛𝑥 is equal to 𝑥 squared plus seven 𝑥 divided by six [𝑥] squared plus 25𝑥 plus four divided by six 𝑥 squared minus 𝑥 divided by 36𝑥 squared minus one.

Okay, so we’re gonna look at simplifying this function. Ultimately, to simplify a function like this, what we will end up doing is simplifying the quotient. Okay. So, first of all, we’re gonna use a bit of the one of the rules from when we’re working with fractions. And as you can see, what we’ve actually got is 𝑥 squared plus seven 𝑥 over six 𝑥 squared plus 25𝑥 plus four, now multiplied by 36𝑥 squared minus one over six 𝑥 squared minus 𝑥.

So actually you could see we’ve found the reciprocal, which means the numerator and the denominator have swapped places. Okay, great! So now what do we do? How are we gonna simplify this function? Well as you can see, this type of function, there are a lot of 𝑥 squared values. There’s a lot of 𝑥s that’s actually quadratic in there. So what we’re actually gonna be able to do is any kind of problem like this is we’re gonna factor our numerators and factor our denominators.

Okay, so we’re gonna start with 𝑥 squared plus seven 𝑥. So 𝑥 squared plus seven 𝑥, we have a look there, and the factor that goes into both 𝑥 squared and seven 𝑥 is 𝑥. So 𝑥 would be outside the parentheses. And inside the parenthesis, what we’d look at is well 𝑥 multiplied by 𝑥 will give us 𝑥 squared, and then 𝑥 multiplied by positive seven will give us seven 𝑥. We’re now gonna factor the denominator, okay? So to remind ourselves of what we need to do here, we can see that it’s a quadratic, so therefore we know we’re gonna have a pair of parentheses.

We now need to work out what’s gonna go into the parentheses as factors. As with this quadratic, it’s a little bit more complicated because we’ve actually got a coefficient of 𝑥 squared which is greater than one, so we’ve got coefficient of 𝑥 squared which is six, so we’re just gonna go through a little method that you can use to actually work out what the factors for this quadratic would be. First of all, we’re gonna multiply 𝑎 by 𝑐, so we’re gonna have six times positive four. And now what we need to do is we actually have to find out what two numbers will multiply together to give us our 𝑎𝑐 values, so 24, so positive 24, but add together to give our 𝑏 value of positive 25.

So in this example, we know that actually the numbers 24 and one will work cause 24 times one gives us 24 and 24 add one gives us 25, positive 25. So we now take the 24 and the one and we actually replace them back into our quadratic in place of the 25𝑥 cause we actually split the coefficient of 𝑥 into two parts. Okay, we’re now in a position where actually we can now start to work out the factors because what we actually do is we now factor our quadratic in two sections, so we factor the first two terms and then we factor the second two terms.

So when we factor the first two terms, we can take six as a factor and 𝑥 as factor, so they go outside the parenthesis. And then inside the parenthesis, we have 𝑥 because six 𝑥 multiplied by 𝑥 gives us six 𝑥 squared, and we have plus four because positive four multiplied by six 𝑥 gives us 24𝑥. Our next two terms actually don’t require in this instance any further factoring. It just gives us 𝑥 plus four. For the second part, we do have to take plus one outside the parentheses as our factor, so then that will give us plus one 𝑥 plus four.

Now finally, we can fully factor what we have, and the way we do that is, again, check you should always have in the parentheses the same factor; that’s how you know you get this method correct. And that means we now know our two factors will be: in the first parenthesis, it’s the beginning factor of each of the left- and right-hand factoring that we’ve been doing so six 𝑥 plus one; and the second parentheses, we’ll have 𝑥 plus four. So now we know that our quadratic six 𝑥 squared plus 25𝑥 plus four gives us the factors six 𝑥 plus one and 𝑥 plus four.

Okay, so we’re now going to factor the right-hand side of our function, and we’re to start with the numerator. This is actually a particular type of factoring which is called the difference of two squares. And I’m just going to demonstrate this now. The difference of two squares is a way of factoring a particular type of expression. So we have 36𝑥 squared minus one. What we can actually notice with this is that they’re in fact, all three parts are squared numbers or squared terms; 36 is six squared, 𝑥 squared is 𝑥 squared, and one is one squared. So what you can see is that actually each part of this term is a square.

The other key factor to be able to use the difference of two squares is the fact that there is a minus sign. So you’ve got 36𝑥 squared, so our 𝑥 squared term minus our square number. And this is the key when we’re gonna use the difference of two squares. Well the difference of two squares, we’re gonna have a pair of parentheses. And in front of each of the parentheses, we’re gonna have the root of our 𝑥 squared term, so the root of 36𝑥 squared. So in this case, it’s gonna give us six 𝑥, in each, then the last term in the parentheses is going to be root of one which gives us one in each.

Now the other thing that we need to remember when we’re doing the difference of two squares is that each bracket must be a different sign, so one will be positive, one will be negative. And the reason for this is so that when we multiply out the brackets, we’ll get 36𝑥 squared and we’ll have minus six 𝑥 plus six 𝑥 so they cancel each other out and leave us with the 36𝑥 squared minus one. And also multiplying a positive and a negative will leave us with the negative one that we need.

Okay, that means we’ve now factored the numerator. Now the final thing to do is to factor the denominator of that side. And to do that, we have 𝑥 is a factor of both terms so we take 𝑥 outside the parentheses, which gives us 𝑥 open parentheses six 𝑥 minus one. And the reason is because 𝑥 times six 𝑥 gives us six 𝑥 squared and 𝑥 times negative one gives us minus 𝑥 or negative 𝑥. Fantastic. Great. Now I can multiply the numerators and denominators because it’s multiplication, which gives us the function in this form.

Now a top tip, whenever you’re trying to simplify a function like this, and we’ve factored the numerator and denominator, you will always always always have some factors on the top that are the same as the factors on the bottom. So we’re gonna have some factors in the numerator the same as the factors on the denominator. If you don’t, then please check your answers and check your factoring because without it we won’t be able to simplify.

Okay now for the final stage, to simplify the function, what we need to do is we can actually look at our common factors and divide the numerator and denominator by those common factors. So the first one we’re gonna divide by is we’re gonna divide the numerator and denominator both by 𝑥 as this is a common factor. We can then divide our numerator and our denominator by six 𝑥 plus one because this is a common factor. And then finally, our last common factor is six 𝑥 minus one so we can divide the numerator and the denominator by six 𝑥 minus one, which leaves us with the function equalling 𝑥 plus seven divided by 𝑥 plus four.

Now is this the final answer? Is it simplified fully? Well check that we can’t factor either the numerator or denominator any further. And if we can’t, then yes this is our final simplified answer. Okay, so little recap of what we’ve done to simplify the function. So first of all because it was dividing two fractional functions, we have to find the reciprocal of the second function and then multiply by it. The next stage is check for any quadratics or terms that can be factored, and then we factor anything that can be factored in the numerators and denominators.

We then multiply the numerators and denominators as we would with any fraction, and then finally, our key tip, check for our common factors. And once we’ve got our common factors, we divide the numerator and denominator by our common factors to leave us with our simplified function, remembering to check that it is simplified as far as it can go and neither numerator or denominator can be simplified or factored any further.

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