Lesson Video: Multiplying and Dividing Rational Expressions Mathematics

Video Transcript

In this lesson, what we’ll learn to do is how to multiply and divide rational expressions. So, the objectives we’ll be looking at are how to multiply rational expressions, expressing amounts in its simplest form, so that might mean that we’re going to factor the numerator and denominator, if necessary. Then, we’re also gonna look at dividing rational expressions, expressing an answer in its simplest form. And we’ll do this with expressions with two terms and then finally finish up with one with three terms.

So before we start with looking at some examples, we firstly think, “Well, what is a rational expression?” Well, with rational expressions, there are a few definitions we can look at. The first of them being the quotient of two polynomials, which we can think of in a more simple way as a fraction whose numerator and denominator are polynomials, like the example we have here: 𝑥 plus four over 𝑥 squared plus five 𝑥 plus four. Finally, another way we could think of it is the ratio of two polynomials.

Okay, so now, we should be pretty clear about what a rational expression is. But in this lesson, we’re gonna look at multiplying and dividing rational expressions. Well, if we’re gonna multiply and divide rational expressions, then we’re going to use exactly same rules we would for any fractions. So, for example, if we have 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑, this is equal to 𝑎𝑐 over 𝑏𝑑. And if we divided fractions, what we’d have is 𝑎 over 𝑏 divided by 𝑐 over 𝑑 is equal to 𝑎 over 𝑏 multiplied by the reciprocal of the second fraction, so 𝑑 over 𝑐, which would give us 𝑎𝑑 over 𝑏𝑐. However, as we are dealing with expressions, what we will be using is a number of different skills that we already know, such as factoring, to help us carry out these operations.

So, now we’ve looked at what a rational expression is and we’ve reminded ourselves how we multiply and divide fractions. We’re gonna have a look at first question.

Simplify 𝑥 plus two over 𝑥 plus three multiplied by two 𝑥 plus six over 𝑥 plus one.

So with a problem like this, we could just multiply the numerators and multiply the denominators. However, what we always do when we get a problem like this is look at them first to see, are there any section, so numerators or denominators, that can be factored? Well, on inspection, we can see that the numerator from the right-hand side can in fact be factored. So, what we’re gonna do is factor this first. Well, what we can see is that with our numerator on the right-hand side, we can take out two as a factor. When we do, we’ll have two multiplied by 𝑥 plus three. So therefore, what we’ve got now in our expression is 𝑥 plus two over 𝑥 plus three multiplied by two multiplied by 𝑥 plus three over 𝑥 plus one.

So, in these types of questions, what we’re always looking to do is to divide through by any common factors. So, these common factors will usually be apparent once we’ve done any factoring. And this is the case in this question cause, as you can see, on the left-hand side, the denominator, we’ve got 𝑥 plus three. And now, on the numerator on the right-hand side, we also have the factor 𝑥 plus three. So, what we now can do is divide through by 𝑥 plus three. So now, using the standard rules for multiplying fractions, what we’re gonna have is 𝑥 plus two multiplied by two for our numerator, so that gives us two, multiplied by 𝑥 plus two. And then for our denominator, we’ve got one multiplied by 𝑥 plus one, which just gives us 𝑥 plus one.

Now, we could leave it in this form. But what we’re gonna do is one more step before we give our answer. And that is to distribute across our parentheses. And then if we distribute across our parentheses, we have two multiplied by 𝑥 and two multiplied by two. So, therefore, we can say that if we simplify 𝑥 plus two over 𝑥 plus three multiplied by two 𝑥 plus six over 𝑥 plus one, the result is two 𝑥 plus four over 𝑥 plus one.

Okay, so in this question, what we looked at was multiplying two rational expressions. Now, we’re gonna take a look at an example where we’re going to divide two rational expressions.

Simplify two 𝑥 over 𝑥 plus three divided by two 𝑥 over five 𝑥 plus 15.

So to solve this problem, the first thing we need to do is remind ourselves of how we divide fractions. So, if we have 𝑎 over 𝑏 divided by 𝑐 over 𝑑, this is equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐, so the reciprocal of the second fraction, which is equal to 𝑎𝑑 over 𝑏𝑐. You might also remember the skill using the phrase “keep it, change it, flip it.” Keep the first fraction, change the sign to multiply, and flip the second fraction. So, in our question, what this means is that our expression is gonna become two 𝑥 over 𝑥 plus three multiplied by five 𝑥 plus 15 over two 𝑥. And that’s because that’s reciprocal of two 𝑥 over five 𝑥 plus 15.

So, the next thing we always look to do in this type of problem is take a look at numerators and denominators and see if any of them could be factored. And in fact, yes, in this problem we can see that the right-hand side numerator can be factored. And if we factor five 𝑥 plus 15, well, what we can do is take out five as a factor. So, when we do that, we’ll have five multiplied by 𝑥 plus three. So now our expression is two 𝑥 over 𝑥 plus three multiplied by five multiplied by 𝑥 plus three over two 𝑥. Now, the purpose of this factoring is to see if we can uncover any common factors in the numerator and denominator. And here we have because we have 𝑥 plus three on the left-hand side is the denominator. And we have 𝑥 plus three on the right-hand side in our numerator. So, what this means is that we can divide through by that common factor, as we have done here.

But also, if we take a look at this question, we can also see another common factor. And that common factor is two 𝑥 because on the numerator on the left-hand side, we have two 𝑥 and on the denominator on the right-hand side we have two 𝑥. So, we divide through by this as well. So now, what we’re left with is one over one multiplied by five multiplied by one over one which neatly leaves us with a result of five.

Okay, so we’ve now looked at multiplying and dividing our rational expressions. We’ll now then look at another example of dividing rational expressions. And in this example, our numerators and denominators are already going to be in factored form.

Simplify 24 over 12 multiplied by 𝑛 plus five divided by 𝑛 plus three over 𝑛 plus five multiplied by 𝑛 plus three.

So, the first thing we do is remind ourselves of how we divide fractions. And to do that, what we have is 𝑎 over 𝑏 divided by 𝑐 over 𝑑 is equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐, so the reciprocal of the second fraction, which is equal to 𝑎𝑑 over 𝑏𝑐. So, what we’ve got here is two rational expressions, and we’re gonna divide them. So, using our rule for dividing fractions, what we’re gonna get is 24 over 12 multiplied by 𝑛 plus five multiplied by 𝑛 plus five multiplied by 𝑛 plus three over 𝑛 plus three.

So, in a problem like this, before we carry out any multiplying of numerators and denominators, what I like to do is see if we got any common factors that we can divide through by. But we can see that this is the case because we’ve got 𝑛 plus five is a common factor in the left-hand denominator and the right-hand numerator. And then if we look at the right-hand expression, we’ve got a common factor of 𝑛 plus three on the numerator and dominator. So first of all, we can divide through by 𝑛 plus five, which cancel these out, and then we can divide through by 𝑛 plus three. So now, what we can do is work out the result. But what we’re left with is 24 over 12 multiplied by one multiplied by one multiplied by one over one. Well, the right-hand expression is just one. So, we’ve got 24 over 12 multiplied by one, which is just equal to two.

Okay, great. So, we’ve now looked at an example which was already in fully factored form. Now, what we’re gonna do is look at how we’re going to multiply two rational expressions. But this time, we’re gonna have to do some factoring of a quadratic.

Simplify 𝑥 plus four over 𝑥 minus one multiplied by 𝑥 plus one over 𝑥 squared plus three 𝑥 minus four.

So, whenever we’re trying to solve this type of problem, before we look at multiplying numerators and multiplying denominators, what we want to do is see if any of the terms that we’ve got can be factored. Well, in fact, we can see yes, there’s one that can be because the denominator from the right-hand side is a quadratic. Well, we think it can be factored. But let’s check. Because even though it’s a quadratic, this doesn’t automatically mean it can be factored. So, we’ve got 𝑥 squared plus three 𝑥 minus four.

Now, to remind ourselves what we do if we want to factor quadratic, what we’re looking for is two numbers whose product would give us, in this case, the negative four and whose sum would give us positive three. Well, the two values we need would be four and negative one. That’s because four multiplied by negative one is negative four and four add negative one is positive three. So therefore, we know that this can be factored. So, 𝑥 squared plus three 𝑥 minus four becomes 𝑥 plus four multiplied by 𝑥 minus one.

So now, what we can do is substitute this back into our expression. So when we do that, what we have is 𝑥 plus four over 𝑥 minus one multiplied by 𝑥 plus one over 𝑥 plus four multiplied by 𝑥 minus one. Now the reason why we do this while we’re factoring a thing that can be factored is to search for any common factors on the numerators and denominators. And now, once we’ve done this, we can see that we’ve got one because we’ve got 𝑥 plus four in the numerator of the left-hand side and 𝑥 plus four in the denominator of the right-hand term. So now, what we can do is divide through by this factor, which we’ve done. So now, what we’ve got is one over 𝑥 minus one multiplied by 𝑥 plus one over one multiplied by 𝑥 minus one.

So, now, what we need to do is multiply the numerators and multiply the denominators. So, when we do that, we’re left with our fully simplified answer, which is 𝑥 plus one over 𝑥 minus one squared. And we got 𝑥 minus one squared in the denominator because we had 𝑥 minus one multiplied by 𝑥 minus one.

Okay, so for our last problem, what we’re gonna take a look at is a more complex example. This one will have three terms, and it will also include multiplication and division.

Simplify 𝑥 plus five over 𝑥 plus six divided by 𝑥 squared plus 10𝑥 plus 25 over 𝑥 squared plus six 𝑥 multiplied by six 𝑥 over 𝑥 squared plus 11𝑥 plus 30.

So what we can see is that in this problem, we’ve got three expressions or terms, and they are three rational expressions. So, what we’re going to do is divide and multiply them. And what we’re gonna start with is the bit inside the parentheses. Because if we think about our order of operations, PEMDAS, then we know that parentheses come first. So therefore, what we’re gonna do is take a look at 𝑥 squared plus 10𝑥 plus 25 over 𝑥 squared plus six 𝑥 multiplied by six 𝑥 over 𝑥 squared plus 11𝑥 plus 30.

Well, the first thing we do with anything like this is we take a look at the numerators and dominators to see if any of them could be factored. And here, we’ve identified three. So now, what we want to do is factor each of them. But if we take a look at the numerator on the left-hand side, this is a quadratic, and this will factor to 𝑥 plus five multiplied by 𝑥 plus five or 𝑥 plus five squared. And the reason we know that this is the case is because we’re looking for two values that whose product is positive 25 and whose sum is positive 10. And we know that five multiplied by five is 25 and five add five is 10. Okay, great. So that’s the numerator on the left-hand side factored.

Now, if we take a look at the denominator on the left-hand side, we can see that a common factor here is 𝑥. So, we can take 𝑥 outside the parentheses. So, what we’ve got is 𝑥 multiplied by 𝑥 plus six. So then, if we take a look at the numerator of the right-hand term, this is six 𝑥 because it can’t be factored. And then what we’ll do is take a look at the dominator. Well, once again, it’s a quadratic. So, we want the product of the two numbers to be 30 and the sum to be positive 11. So therefore, what we’re gonna have is the factored form is 𝑥 plus six multiplied by 𝑥 plus five. That’s cause six multiplied by five is 30, six add five is 11. So now, what we have is our expressions in fully factored form.

Okay, great. So now, what’s the next step? Well, the next step is to see if there’re any common factors in the numerators and denominators. Well, we can see that 𝑥 plus five is a common factor cause it’s in the numerator of the left-hand term and the denominator of the right-hand term. So now what we do is we take a look to see if there are any other factors. And we can see, well, yes, there’s an 𝑥 on the right-hand numerator and an 𝑥 on the left-hand denominator. So, we can divide through by 𝑥. So now, what we can do is simplify what we’re left behind with. And what that will give us is six multiplied by 𝑥 plus five over 𝑥 plus six squared or 𝑥 plus six multiplied by 𝑥 plus six. Now, this certainly could have been written as 𝑥 plus six squared on the denominator. But I’ve just kept it in this format so that it’s gonna be easier to cancel in a minute.

So now, we deal with the parenthesis. What we’re gonna do is go back to the original calculation. So now what we’ve got is 𝑥 plus five over 𝑥 plus six divided by six multiplied by 𝑥 plus five over 𝑥 plus six squared. So now to carry out this calculation, we just need to remind ourselves about how we divide a fraction. So that’s 𝑎 over 𝑏 divided by 𝑐 over 𝑑 is equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐, which is the same as 𝑎𝑑 over 𝑏𝑐. So what we do is we multiply by the reciprocal of the second fraction. So what we do now is we apply this rule. And we get 𝑥 plus five over 𝑥 plus six multiplied by 𝑥 plus six squared over six multiplied by 𝑥 plus five.

So now as we did before, we identify any common factors and we divide through by these. So we can see we’ve got 𝑥 plus five and 𝑥 plus six as common factors. So first of all, we can divide through by 𝑥 plus five and then through by 𝑥 plus six. So now, all we need to do is multiply what we’re left with, so multiply our numerators and multiply our denominators. And when we do this, what we’re left with is 𝑥 plus six over six. And that’s our fully simplified form because neither the numerator nor the denominator can be simplified any further.

Okay, great. So we’ve looked at a range of problems. And they’ve included multiplying and dividing rational expressions; they’ve included factoring. And we’ve even looked at an example that included both multiplying and dividing because it had three expressions. So now, what we’re gonna do is take a look at the key points of the lesson. So, our first key point was to have a look at what a rational expression was. And a rational expression is a quotient of two polynomials or, more simply put, is a fraction whose numerator and denominator are polynomials.

We also reminded ourselves how we actually multiply and divide fractions because this is the same when we’re dealing with rational expressions. So, for example, 𝑎 over 𝑏 multiplied by 𝑐 over 𝑑 is equal to 𝑎𝑐 over 𝑏𝑑 and 𝑎 over 𝑏 divided by 𝑐 over 𝑑 is equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐. So, we multiply by the reciprocal of the second fraction, which is equal to 𝑎𝑑 over 𝑏𝑐.

And finally, to remind us the method that we used, if we’re multiplying two rational expressions, first of all, we factor where possible. So, we look at numerator and dominator to see if there’re any terms that can be factored. Then, we identify any common factors and divide through by these common factors and then simplify what’s left. So, we multiply the numerators and denominators. The only difference with this when we’re doing division is that we, first of all, carry out like we said before the rule for division, which means that we flip the second fraction or find the reciprocal and then we carry out our multiplication, given these four steps.