### Video Transcript

Okay, weâ€™re going to fully simplify this expression that we have here in front of us. So we have đť‘Ą squared plus three đť‘Ą plus two divided by đť‘Ą squared plus four đť‘Ą plus three.

At first glance, we look at this and we think, â€śRight! Well where where do we begin?â€ť You think, â€śOh! Can we divide three by đť‘Ą? Are there any numbers that will go into these as a factor.â€ťAnd we canâ€™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 itâ€™s asked you to fully simplify, the chances are youâ€™re gonna be able to factor them. So youâ€™re gonna factor the numerator and factor the denominator, so letâ€™s have a go and see if we can do that.

Weâ€™re gonna start by factoring the numerator. Well as itâ€™s a quadratic, we know that weâ€™re gonna have the pair of parentheses, which weâ€™ll put in now. We also know that as itâ€™s an đť‘Ą squared and the coefficient of the đť‘Ą squared is one, weâ€™re gonna have an đť‘Ą 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 đť‘Ą 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 Iâ€™ve put in green, something to help us when we are factoring. So we want to find the coefficient of đť‘Ą. The way to find the coefficient of đť‘Ą, the two factors that are in our parentheses have to add together to give us the coefficient of đť‘Ą and they have to multiply together to give us the integer at the end of our quadratic.

So letâ€™s have a look at the numerator. Okay so we can see with the numerator, in the parentheses, our factors, theyâ€™re going to need to add together to make positive three, because itâ€™s a coefficient of đť‘Ą, and multiply together to make positive two because thatâ€™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! Weâ€™ve factored our numerator. Now weâ€™re gonna have a go at factoring the denominator. Again, weâ€™ve got a pair of parentheses with đť‘Ą at the beginning of each, but now we obviously need to find our factors in each of these parentheses. For the denominator, theyâ€™re gonna have to add to make positive four and multiply to make positive three. So we get đť‘Ą plus three and đť‘Ą plus one. Again, three times one makes three, and three plus one makes four.

Fantastic! Weâ€™ve now factored the numerator and denominator. At this point, Iâ€™m going to introduce you a key tip. Obviously this type of question, theyâ€™re gonna want you to factor. And if you do factor, the reason theyâ€™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 isnâ€™t a common factor, you canâ€™t see one, in that case what Iâ€™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 weâ€™re at a point where we can fully simplify. Right! So what weâ€™d notice is that our numerator and our denominator both contain the factor đť‘Ą plus one. So what weâ€™re going to do is weâ€™re going to divide the numerator and divide the denominator by đť‘Ą plus one. Fantastic! Weâ€™re not at a stage where weâ€™ve got đť‘Ą plus two divided by đť‘Ą 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! Weâ€™ve fully simplified the expression and I just wanna quickly recap what weâ€™ve done.

So first stage, have a look at your expression, think,â€śRight. What have we got?â€ť Okay, if weâ€™ve got quadratics or weâ€™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 thereâ€™s a common factor.

Once you find the common factor, divide the numerator and the denominator by that common factor, and then youâ€™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.