Fully simplify 𝑥 squared minus four over 𝑥 squared minus four 𝑥 plus four.
We simplify algebraic fractions the same way we simplify any numerical fraction. We divide both the numerator and the denominator of that fraction by the greatest or the highest common factor. To find this for algebraic fractions though, that requires a little bit more work. And what we do is we begin by factoring the expressions on the numerator and the denominator of our fraction.
We’ll factor 𝑥 squared minus four. And you might’ve noticed that this is this is a special type of quadratic expression. We have two square numbers, 𝑥 squared and four. And they’re separated by a subtraction symbol. This is known as the difference of two squares, sometimes written as DOTS. And for the general expression 𝑎 squared minus 𝑏 squared, we can write that as 𝑎 plus 𝑏 multiplied by 𝑎 minus 𝑏.
So, in this example, we see that the number at the front, or the letter at the front, of each bracket is 𝑥. And we see that the square root of four is two. So, we have a two as our numerical part in each bracket. And we can see that 𝑥 squared minus four factors to 𝑥 plus two multiplied by 𝑥 minus two. It’s always sensible to expand this back out or distribute these brackets again and check we get the same answer. And we can do that using the FOIL. method.
F stands for first. We multiply the first term in each bracket. 𝑥 multiplied by 𝑥 is 𝑥 squared. O stands for outer. We multiply the outer two terms. That’s 𝑥 multiplied by negative two, which is negative two 𝑥. We then multiply the inner terms. That’s two 𝑥. And L stands for last. Two multiplied by negative two is negative four. And we see that negative two 𝑥 plus two 𝑥 gives us zero. And this simplifies to 𝑥 squared minus four, which was our original expression. And so, we factored 𝑥 squared minus four successfully.
Next, we repeat this process for the denominator. Except this time, this expression is not the difference of two squares. We do know that it will factor into two brackets. There are no common factors in each term apart from one. And we also know that an 𝑥 must go at the front of each bracket, since the first thing that we do is multiply the first term in each bracket. And here we would get 𝑥 squared by multiplying 𝑥 by 𝑥.
To find the numerical part for each bracket though, we need to look for two numbers that multiply to make this constant, four, and add to make the coefficient of 𝑥. That’s negative four. We always start by listing the factors of four. And it might seem as though one and four and two and two are the only factor pairs for the number four. But remember a negative times a negative is a positive. So, we also have negative one multiplied by negative four and negative two multiplied by negative two to give us four.
The only factor pair that sums to make negative four is negative two and negative two. So, this expression factors to be 𝑥 minus two multiplied by 𝑥 minus two. And we could even write this as 𝑥 minus two squared, though it’s not actually necessary in this question.
And so, we can write out our algebraic fraction as 𝑥 plus two multiplied by 𝑥 minus two over 𝑥 minus two multiplied by 𝑥 minus two. And this is great! We can see that there’s a common factor in both the numerator and the denominator. It’s 𝑥 minus two. So, we divide through by 𝑥 minus two. And we’re left with 𝑥 plus two over 𝑥 minus two. And 𝑥 plus two and 𝑥 minus two are what we call coprime. They share no common factors aside from one. And this means that this fraction is fully simplified. It’s 𝑥 plus two over 𝑥 minus two.
Now at this point, it’s useful to know that if you’re struggling to factorise either the numerator or the denominator, it can be useful to look at the brackets in the expression you’ve always factorised. There’s a good chance that they’ll have one in common and you can use that as a starting point.