Lesson Video: Simplifying Algebraic Fractions Mathematics

Video Transcript

In this video, we’ll learn how to factor algebraic expressions and simplify algebraic fractions. Let’s begin by recalling the general method for simplifying a fraction made up of purely numerical values, such as 27 over 63. Our job is to find the greatest common factor of the numerator and denominator of our fraction, then divide through by this value.

So take the fraction 27 over 63. Let’s find the greatest common factor of these numbers. The greatest common factor of 27 and 63 is nine. So we divide the numerator and denominator of our fraction by nine, giving us an answer of three-sevenths. We say that three and seven are coprime. They share no factors other than one. So we can say that our fraction is now fully simplified. And believe it or not, the same process goes for algebraic fractions.

Let’s take five 𝑥 squared over 10𝑥 to the fifth power. This time, we need to work out the greatest common factor of five 𝑥 squared and 10𝑥 to the fifth power. Well, the greatest common factor of the numerical part that is the greatest common factor of five and 10 is five and, of our algebraic part, it’s 𝑥 squared. So the greatest common factor of our numerator and denominator is five 𝑥 squared. We then divide both parts of this fraction by five 𝑥 squared. Five 𝑥 squared divided by five 𝑥 squared is one. And 10𝑥 to the fifth power divided by five 𝑥 squared is two 𝑥 cubed. So in its simplest form, our fraction is one over two 𝑥 cubed. But what if our algebraic fraction is more complicated? Let’s see what that might look like.

Fully simplify 𝑥 plus four times 𝑥 plus three over 𝑥 plus two times 𝑥 plus four.

Remember, to simplify fractions, we divide the numerator and denominator by the greatest common factor of each. Let’s look carefully at the numerator and denominator of our fraction. The factors of the numerator are 𝑥 plus four and 𝑥 plus three. Then the factors of the denominator are 𝑥 plus two and 𝑥 plus four. We can therefore say that the greatest common factor is 𝑥 plus four. It’s the only factor other than one that these two expressions have in common.

So to simplify this algebraic fraction, we’re going to divide through by 𝑥 plus four on both parts. 𝑥 plus four times 𝑥 plus three divided by 𝑥 plus four leaves us with 𝑥 plus three on our numerator. Similarly, 𝑥 plus two times 𝑥 plus four divided by 𝑥 plus four leaves us with 𝑥 plus two on our denominator. That leaves us with 𝑥 plus three over 𝑥 plus two. 𝑥 plus three and 𝑥 plus two are now coprime. They share no other factors other than one. So we’re finished. In its simplest form, our fraction is 𝑥 plus three over 𝑥 plus two.

Now, this fraction was quite easy to simplify because it was fully factored on both its numerator and denominator. We’ll now have a look at an example where that’s not necessarily the case.

Fully simplify 𝑥 minus three times 𝑥 squared minus six 𝑥 plus nine over 𝑥 minus three cubed.

Remember, to simplify fractions, we divide the numerator and denominator by the greatest common factor of each. Now, we can see that we have a common factor of 𝑥 minus three. There’s an 𝑥 minus three in the numerator and the denominator. But is that the greatest common factor? Well, to check, we factor any nonfactored expressions. So we’re going to fully factor the expression 𝑥 squared minus six 𝑥 plus nine from the numerator of our fraction. This is a quadratic expression, but there are no common factors in 𝑥 squared, negative six 𝑥, and nine. So that tells us we factor into two parentheses.

We know that, in order to achieve an 𝑥 squared, we need an 𝑥 here and an 𝑥 here. And we need to find two numbers whose product is nine and whose sum is negative six. Well, that must be negative three and negative three, since negative three times negative three is positive nine. But negative three plus negative three is negative six. If we replace 𝑥 squared minus six 𝑥 plus nine with its factored form, our fraction becomes 𝑥 minus three times 𝑥 minus three times 𝑥 minus three over 𝑥 minus three cubed. But of course, it should be quite clear that 𝑥 minus three times 𝑥 minus three times 𝑥 minus three is actually 𝑥 minus three cubed.

Notice now that our numerator and denominator are actually equal. We’re dividing a number by itself. And when we divide a number by itself, we get one. So in simplified form, this fraction is simply one. Now, note that we could’ve approached this slightly differently. Instead of factoring at the start 𝑥 squared minus six 𝑥 plus nine, we could have divided through by a factor of 𝑥 minus three. We eventually saw that this isn’t the greatest common factor, but it’s a good start. We divide the numerator and the denominator by 𝑥 minus three, noting that 𝑥 minus three cubed divided by 𝑥 minus three is 𝑥 minus three squared.

Then we could’ve factored and spotted that we had further common factors of 𝑥 minus three squared. Either method ultimately results in us dividing both the numerator and denominator by the greatest common factor of 𝑥 minus three cubed. And either method results in an answer of one.

Let’s now have a look at an example that involves factoring more than one expression.

Fully simplify 𝑥 plus two times 𝑥 squared plus seven 𝑥 plus 12 over 𝑥 plus seven times 𝑥 squared plus 10𝑥 plus 21.

Remember, to simplify fractions, we divide the numerator and denominator by their greatest common factor. Now, looking at our fraction, it’s not instantly obvious what the greatest common factor is. And so we look for any unfactored expressions and we factor them. In fact, there are two. On the numerator, we have 𝑥 squared plus seven 𝑥 plus 12 and, on the denominator, 𝑥 squared plus 10𝑥 plus 21. Note that 12 and 21 share a common factor of three. That’s an indication to us that we might end up canceling by a factor of, say, 𝑥 plus three. That’s not hugely helpful just yet, but certainly something to bear in mind.

We begin by fully factoring the expression 𝑥 squared plus seven 𝑥 plus 12. It’s a quadratic expression. And there are no common factors throughout. So we have two brackets with an 𝑥 at front of each bracket. We’re looking for two numbers that multiply to make 12 and add to make seven. Well, that’s three and four. So 𝑥 squared plus seven 𝑥 plus 12 can be written as 𝑥 plus three times 𝑥 plus four. Let’s repeat this process for the expression 𝑥 squared plus 10𝑥 plus 21.

Once again, it’s two pairs of parentheses with the 𝑥 at the front of each one. This time, though, we want two numbers whose product is 21 and whose sum is 10. That’s seven and three. So this expression becomes 𝑥 plus seven times 𝑥 plus three. Let’s now go back to our original fraction and replace each expression with its factored form. In doing so, it becomes 𝑥 plus two times 𝑥 plus three times 𝑥 plus four over 𝑥 plus seven times 𝑥 plus seven times 𝑥 plus three. We’re now ready to look for the greatest common factor of both parts of our fraction. If we look carefully, we see that each part shares a factor of 𝑥 plus three.

So we’re going to divide both the numerator and the denominator by 𝑥 plus three. Dividing the numerator by 𝑥 plus three and we leave ourselves with 𝑥 plus two times 𝑥 plus four. Similarly, dividing the denominator by 𝑥 plus three leaves us with 𝑥 plus seven times 𝑥 plus seven. Noting, of course, that we can write 𝑥 plus seven times itself as 𝑥 plus seven squared, we see that this simplifies to 𝑥 plus two times 𝑥 plus four over 𝑥 plus seven squared. Remember also that multiplication is commutative. We can write our numerator alternatively as 𝑥 plus four times 𝑥 plus two. And we’re still getting the same result. We’ve fully simplified our original fraction.

In our next example, we’ll look at a fraction whose numerator and denominator are both fully unfactored.

Fully simplify 𝑥 squared plus five 𝑥 minus 24 over 𝑥 squared plus 15𝑥 plus 56.

We begin by recalling that, to simplify fractions, we divide the numerator and denominator by their greatest common factor. Now, it’s not instantly obvious what the greatest common factor of 𝑥 squared plus five 𝑥 minus 24 and 𝑥 squared plus 15𝑥 plus 56 is. And so we look for any unfactored factorable expressions, and we fully factor them. So let’s begin by factoring 𝑥 squared plus five 𝑥 minus 24. We have a quadratic expression whose three terms are coprime. That is, their greatest common factor is one. That tells us we’re going to factor the expression into two pairs of parentheses, at the front of which will be 𝑥.

Then to find the numerical parts, we need two numbers whose product is negative 24 and whose sum is five. Those are negative three and eight. So 𝑥 squared plus five 𝑥 minus 24 can be written as 𝑥 minus three times 𝑥 plus eight. Our next job is to factor the denominator, 𝑥 squared plus five 𝑥 plus 56. We might notice that 56 has a factor of eight. That’s an indication to us that we might end up dividing through by a factor of 𝑥 plus eight at the end. Let’s factor this expression. Once again, we have two pairs of parentheses. This time we want two numbers whose product is positive 56 and whose sum is positive 15. Well, that’s seven and eight. So this factors to 𝑥 plus seven times 𝑥 plus eight.

And so we rewrite our fraction completely. We write it as 𝑥 minus three times 𝑥 plus eight over 𝑥 plus seven times 𝑥 plus eight. And now, we see that the greatest common factor of 𝑥 minus three times 𝑥 plus eight and 𝑥 plus seven times 𝑥 plus eight is indeed 𝑥 plus eight. And so let’s divide both the numerator and denominator by this value. Dividing our numerator by 𝑥 plus eight leaves us with 𝑥 minus three. Similarly, dividing our denominator by 𝑥 plus eight leaves us with 𝑥 plus seven. And so we fully simplified our fraction. It’s 𝑥 minus three over 𝑥 plus seven.

In our final example, we’ll look at how we can fully simplify an algebraic fraction where either the numerator or the denominator’s coefficient of 𝑥 squared is not equal to one.

Fully simplify 𝑥 squared minus 𝑥 minus 20 over two 𝑥 squared plus nine 𝑥 plus four.

We know that, to simplify any fraction, we need to divide both the numerator and the denominator by their greatest common factor. The problem is, with algebraic fractions, it’s not instantly obvious what that is. And so before we can divide through by the greatest common factor, we look to fully factor any unfactorized factorable expressions. We actually have two. We have the numerator, 𝑥 squared minus 𝑥 minus 20, and then the denominator, two 𝑥 squared plus nine 𝑥 plus four. Let’s factor 𝑥 squared minus 𝑥 minus 20. This is a quadratic expression whose three terms are coprime. They share no other factors than one.

This tells us we factor into two pairs of parentheses with an 𝑥 at the front of each. Now, we need to find two numbers whose product is negative 20 and whose sum is negative one. That’s negative five and plus four. So we fully factored the numerator. But what about the denominator? This one is a little trickier since the coefficient of 𝑥 squared is no longer one, it’s two. This tells us that, in one of our parentheses, we’re going to have two 𝑥 at the front. And so we can’t use the standard tricks. We do know we need two numbers whose product is four. Well, this could be one and four, which means that we could have four and one in our parentheses. Alternatively, we could have two and two.

Now, we could use a bit of trial and error, but let’s look carefully at our numerator. One of the factors of our numerator is 𝑥 plus four. That’s a good indication to us that our denominator must share that factor, meaning that it could factor to be two 𝑥 plus one times 𝑥 plus four. Let’s check by redistributing our parentheses. We multiply two 𝑥 by 𝑥 to give us two 𝑥 squared. We then multiply two 𝑥 by four to get eight 𝑥. We multiply one by 𝑥 to get 𝑥. And then we multiply one by four to get four. Collecting like terms, and we see that we do indeed get two 𝑥 squared plus nine 𝑥 plus four as required.

And so we see we can write our fraction as 𝑥 minus five times 𝑥 plus four over two 𝑥 plus one times 𝑥 plus four. The greatest common factor here then must be 𝑥 plus four. So we’ll divide both the numerator and the denominator by 𝑥 plus four. Dividing our numerator leaves us with 𝑥 minus five and dividing our denominator leaves us with two 𝑥 plus one. And so we fully simplified our algebraic fraction. It’s 𝑥 minus five over two 𝑥 plus one.

Now, it is worth going back to this step here, where we factorized two 𝑥 squared plus nine 𝑥 plus four. We made an assumption that our algebraic fraction was going to simplify and that there must be a common fact of 𝑥 plus four. It might have been, however, but this was not how two 𝑥 square plus nine 𝑥 plus four should have been factored. If that had been the case and we hadn’t ended up with some common factor, then we could’ve deduced that our fraction was already in its simplest form.

In this video, we learned that we can simplify algebraic fractions the same way we do numerical fractions, by dividing both the numerator and the denominator by their greatest common factor. We also saw that there will be occasions where the greatest common factor isn’t instantly obvious. In those occasions, we’ll need to factor any unfactored expressions before it becomes apparent.