Okay, we’re going to fully simplify this expression that we have here in front of us. So we have 𝑥 squared plus three 𝑥 plus two divided by 𝑥 squared plus four 𝑥 plus three.
At first glance, we look at this and we think, “Right! Well where where do we begin?” You think, “Oh! Can we divide three by 𝑥? Are there any numbers that will go into these as a factor.”And we can’t see anything straight away, but what we can notice is the numerator and the denominator are both in fact quadratics. When you have a question that has the numerator and the denominator as quadratics and it’s asked you to fully simplify, the chances are you’re gonna be able to factor them. So you’re gonna factor the numerator and factor the denominator, so let’s have a go and see if we can do that.
We’re gonna start by factoring the numerator. Well as it’s a quadratic, we know that we’re gonna have the pair of parentheses, which we’ll put in now. We also know that as it’s an 𝑥 squared and the coefficient of the 𝑥 squared is one, we’re gonna have an 𝑥 at the beginning of each of our parentheses. But what about the other two numbers? Which numbers are we actually gonna have within the parentheses?
And there, we have to think back to when we factored in the past, what do we need. So how do we find the coefficient of 𝑥 and how do we found the coefficient of the integer at the end of our quadratic?
As a little reminder, on the right-hand side I’ve put in green, something to help us when we are factoring. So we want to find the coefficient of 𝑥. The way to find the coefficient of 𝑥, the two factors that are in our parentheses have to add together to give us the coefficient of 𝑥 and they have to multiply together to give us the integer at the end of our quadratic.
So let’s have a look at the numerator. Okay so we can see with the numerator, in the parentheses, our factors, they’re going to need to add together to make positive three, because it’s a coefficient of 𝑥, and multiply together to make positive two because that’s the end of our quadratic. We can see that two numbers that will fit that are positive two and positive one because two times one is two and two plus one is three.
So great! We’ve factored our numerator. Now we’re gonna have a go at factoring the denominator. Again, we’ve got a pair of parentheses with 𝑥 at the beginning of each, but now we obviously need to find our factors in each of these parentheses. For the denominator, they’re gonna have to add to make positive four and multiply to make positive three. So we get 𝑥 plus three and 𝑥 plus one. Again, three times one makes three, and three plus one makes four.
Fantastic! We’ve now factored the numerator and denominator. At this point, I’m going to introduce you a key tip. Obviously this type of question, they’re gonna want you to factor. And if you do factor, the reason they’re getting you to do that is so you can simplify. And in order to do that, you need to check for a common factor. If there isn’t a common factor, you can’t see one, in that case what I’d recommend is you check your factor and make sure that you factored correctly because it may be the case that you made an error there because there should always be a common factor.
Now at we’re at a point where we can fully simplify. Right! So what we’d notice is that our numerator and our denominator both contain the factor 𝑥 plus one. So what we’re going to do is we’re going to divide the numerator and divide the denominator by 𝑥 plus one. Fantastic! We’re not at a stage where we’ve got 𝑥 plus two divided by 𝑥 plus three.
Right! Is this fully simplified? Well neither of our terms can factor, so that does mean that we have reached the end and we have fully simplified this expression. Great! We’ve fully simplified the expression and I just wanna quickly recap what we’ve done.
So first stage, have a look at your expression, think,“Right. What have we got?” Okay, if we’ve got quadratics or we’ve got other expression terms, can they factor? Can we factor them? And the answer is, in this case, yes. Then you factor the numerator, factor the denominator, then, remember my tip, check for that common factor. Make sure you see if there’s a common factor.
Once you find the common factor, divide the numerator and the denominator by that common factor, and then you’ll get to stage we should have a fully simplified expression. But again, remember, when you get to that last stage, check to make sure it cannot be simplified any further.