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
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
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
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.