Algebra 2 Multiplying and Dividing Rational Expressions

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Today, we are going to start a series of lectures on rational expressions and equations, starting out with multiplying and dividing rational expressions.0002

And this may be review from a topic you have had before; but we are going to go ahead and go over it,0011

and then go on to some more advanced topics with rational expressions.0017

I am beginning with simplifying rational expressions, and the definition, first, of a rational expression.0022

A rational expression is actually the ratio of two polynomial expressions.0030

So, it is a quotient; in this case, it is a quotient whose numerator and denominator are polynomials.0035

So, you can just look at this as an algebraic fraction.0042

Consider an example such as 4x divided by (3x - 5): this is a rational expression.0048

The numerator is a polynomial, and the denominator is a polynomial, and it is an algebraic fraction.0057

In algebra, we can think of polynomials as the integers, and then rational expressions as fractions.0064

Another example might be something like (2x² - x + 9), over (x² - 7).0072

Now, recall that, when a fraction is simplified, it has no common factor between the numerator and the denominator, other than 1 or -1.0084

So, let's start out by reviewing how this would work with numbers, and then applying these same concepts to rational expressions.0094

If I look at a fraction such as 10/15, you will immediately recognize that this is not in simplest form.0101

And you might even simplify it without really thinking about the exact steps you are taking.0110

Let's go through those steps, so we can apply them here.0115

If I go ahead and factor this out, I will end up with (5 times 2) over (5 times 3).0119

And what I am going to end up doing is canceling out that common factor to end up with 2/3.0128

And this is in simplest form, because now these numbers, 2 and 3, no longer have any common factor, other than 1 or -1.0136

So, it is the same idea here: to simplify a rational expression, you are going to divide the numerator and denominator by their greatest common factor.0146

Let's look at an example, now, involving a rational expression.0154

The first step is to factor: and the numerator here is already factored, and you will recall that0161

x² - 9 is going to factor into (x + 3) (x - 3).0173

And as I look, I see that I do have a common factor; I have a common binomial factor of (x + 3).0185

So, the same way that I canceled out the common factor here, I am going to do the same here.0192

Now, remember that, when I cancel this out, what I am going to end up with is actually 1 in the numerator,0199

because, although we don't write it out, there actually is a 1 here (1 times x + 3).0205

So, this isn't just going to become 0 in the numerator; once I cancel the (x + 3)s out, I am going to end up with 1/(x - 3).0212

Now, this rational expression is in simplest form.0220

I also want you to notice that, even though there are no variables in the numerator, this is still a rational expression.0224

It is still an algebraic fraction; we end up with the rational expression that we started out with, now in simplest form: 1 divided by (x - 3).0230

Sometimes, factoring out a -1 will help to simplify a rational expression.0245

You might look at a rational expression and think, "Well, there are no common factors."0249

But there are some factors that look pretty close, except the signs are off.0254

So, let's look at an example: if I start out with the rational expression (2 - x)/(3x - 6), and I am asked to simplify this,0260

my first step is always going to be to factor; the numerator is already factored;0277

in the denominator, I have a common factor of 3, so I can factor out a 3.0283

That leaves behind an x, and here a -2.0289

Now, to look at this a little bit more easily, and compare what I have in the numerator and the denominator,0294

I am going to start out by rewriting this with the x first.0299

So, it is actually going to be -x + 2; this allows me to more easily compare this with what I have in the denominator, which is 3 times (x - 2).0304

When I compare this binomial factor in the numerator and the denominator, I see that the only difference here0321

is that the x's are opposite signs (so look at it like this: -x and x), and the 2's are also opposite signs.0328

Therefore, if I factor out a -1 from either the numerator or the denominator (one or the other--I am going to pick the numerator),0338

I can end up with what is inside the parentheses--that factor--having the same sign.0349

So, instead of this being 1 times this, I am going to factor out this -1, and I will get -1.0354

So, if I pull the -1 out, I am going to end up with an (x - 2).0362

And I see that I did this correctly, because if I multiply -1 times x, I am going to get a -x back.0366

If I multiply -1 times -2, I will get a + 2 back.0372

I chose to do the numerator; I could have left the numerator alone and factored the -1 out of the denominator, which would have yielded -x + 2.0376

So, here I factored this from the numerator; and this gives me, now, a common factor, (x - 2)/(x - 2).0386

I can then cancel these out; and this simplifies to -1/3.0395

Here, I looked, and I saw that I had what would have been the same factor, except that the signs on the terms were opposite.0401

In that case, factoring out a -1 will allow you to simplify the rational expression that you are working with.0410

Simplifying is tied closely to working with rational expressions in terms of multiplication and division.0420

Let's start out by reviewing the rules that are involved with multiplying and dividing rational expressions,0430

and also thinking back to how it works when you are multiplying fractions that involve numbers.0439

A lot of the concepts are the same.0445

So, if a/b and c/d are rational expressions, then in order to multiply these two (in order to multiply two rational expressions),0448

what you are going to do is multiply the numerators and multiply the denominators, and end up with something in the form ac/bd.0458

Notice that there is the restriction that b cannot equal 0 and d cannot equal 0,0473

because, as usual, we cannot allow the denominator to become 0, because that would be undefined.0478

Let's first talk just about the multiplication; and then we can go on and talk about division.0485

OK, so first, we are just going to talk about multiplying or dividing rational expression that just have terms that are monomials in the numerator or denominator.0495

And then, we will go on to talk about when you actually have a polynomial that is a binomial, trinomial, or greater in the numerator and/or denominator.0503

Let's start out with this example: 4xy³ times x².0514

And that is going to give me...times 12xy⁵, divided by x⁴.0526

In order to multiply this, what I am going to do is actually, first, cancel out any common factors that I might have.0539

So, to make this easier, instead of trying to multiply x² times x⁴, I am just going to go ahead and cancel out common factors.0552

So, I see here that I have an x here, and I have an x² here; they have a common factor of x.0562

Here, I have another x, and this x cancels.0568

Then, I am just going to multiply what is left behind.0572

And this is going to leave me with 4y³ times 12y⁵, divided by x⁴.0577

Now, I multiply what I have left, which is 4 times 12, and that is going to give me 48.0591

And then, recall that multiplying exponential expressions that have the same base0600

means that I am going to add the exponents, so this is going to give me y⁸ divided by x⁴.0606

So, to multiply, you should first cancel out common factors.0613

Then, go on and multiply what is left in the numerator and what is left in the denominator.0618

Looking at division: if I have rational expressions a/b, divided by c/d, I can handle this by turning this into a multiplication problem.0624

And that is the same way that we handle division...0639

Let's say we were asked to divide 3/4 divided by 5/8.0641

I would handle that by rewriting it as 3/4 times the inverse, the reciprocal, of 5/8, which is 8/5.0649

Once I get to there, I just follow my usual rules for multiplication.0659

where I am going to cancel out common factors, and then that becomes the numerator, 3 times 2, divided by the denominator, 5 times 1.0663

This gives me 6/5.0673

I am going to do the same thing here, only I would be working with rational expressions.0675

So, if I have something like 2xy² divided by 4x²y³ times xy divided by 8x²,0680

divided by...if I am being asked to divide these, I am simply going to rewrite this0703

as 2xy² divided by 4x²y³; so I keep that first rational expression the same.0708

And then, I multiply it by its inverse, which is 8x² divided by xy.0716

From there, I am going to proceed, just as I did up there, with multiplication.0725

And I am going to cancel out common factors and then multiply the numerators and denominators of what remains behind.0731

OK, so I have here a common factor of 4; that cancels to 1; this becomes 2.0740

I also see that I have an x² term in the numerator and the denominator; those cancel.0748

I have an x in the numerator and an x in the denominator; those cancel.0755

And then, I have a y in the numerator and in the denominator; those cancel.0760

I have made my life a lot simpler, because instead of multiplying higher powers and larger constants, I have gotten rid of a lot of that.0767

So, now it comes time to multiply what is left behind.0775

Here I have a 2 and a y; all I have left over here is actually the constant 2; in the denominator, I have a 1--0778

I don't need to write that: I can just write the y³.0785

And actually, I have one more common factor left: so 4y/y³.0789

And if you see that, then you can go ahead and simplify at the end, since this common factor wasn't already canceled.0796

Then I am going to go ahead and simplify even further to give me 4 divided by y².0802

So again, division just involves keeping the first rational expression the same, taking the inverse of the second rational expression...0810

and then I went ahead and canceled out the common factors I found,0823

multiplied the numerator and the denominator, and then checked to make sure that this was simplified.0826

And then, I found another common factor, which I went ahead and took care of there.0831

Factoring: if the rational expressions contain polynomials, you actually may need to factor them before a product or quotient can be simplified.0841

So, in the last problem, I showed how I canceled out common factors before I went ahead0850

and did the multiplication of the numerator and the multiplication of the denominator.0855

I want to do the same thing if I am handling polynomials.0859

Only, in this case, in order to do that canceling out, in order to even find what my factors are,0863

I am going to need to probably factor both the numerator and the denominator, if they are not already factored.0868

For example, 2x² - 6x - 20, divided by x² - 49:0876

if I am asked to multiply that by, say, 5x - 35, divided by x² + 4x + 5,0888

last time we were working with monomials, and then I just went ahead and started canceling out my factors;0899

here, I don't even know what my factors are yet.0904

So, I am going to write these in factored form.0906

I can pull the common factor of 2 out from this trinomial to get x² - 3x - 10.0910

x² - 49 I am going to recognize as (x + 7) times (x - 7).0920

5x - 21 I can factor out a 5--that gives me (x - 7) times 5.0929

Here, I actually have...let's see...this is x² + 4x + 5; let's actually make that a 4 right here...(x + 2) times (x + 2).0935

Let's go ahead and make that simpler to work with.0954

Now, I am looking, and I am seeing that my factoring is not done; I can still factor a little bit more up here.0958

This is going to be an x and an x; and this is negative, so I am going to have a positive here and a negative here.0965

And what I am going to look at is the factors of 10: 1 and 10, and 2 and 5.0972

And when one is positive and one is negative, what combination will give me -3?0977

Well, I can see that 1 and 10 are too far apart, so I am going to work with 2 and 5.0982

And I want to end up with a negative number, so I will make the 5, the larger number, negative.0987

And -5 plus 2 is going to give me -3; so I know that the right combination would be to have a 2 here and my 5 right here, with a negative sign.0993

All right, now I see that I have everything factored as far as it can be factored.1012

So, up here, it states that I need to factor these before I can simplify the product or quotient.1021

So, my next step, after I have factored, is to go ahead and do the simplification by canceling out factors that are common to the numerator and denominator.1029

OK, I see that I have an (x + 2) here; I also have an (x + 2) in the denominator.1043

I have an (x - 7) in the numerator and an (x - 7) in the denominator.1050

I have no other common factors, so I am going to multiply what is left in the numerator and in the denominator.1055

I can simplify this a bit more, because 2 times 5 is 10; that is going to give me 10 times (x - 5), divided by (x + 7) (x + 2).1070

And then, I double-check again at the end, to make sure I haven't missed any factors and there are no common factors.1080

It is good practice to simplify, of course, before you multiply, to make your multiplication easier.1086

But then, at the end, go back and double-check and make sure that there are no common factors--that you can't do any more canceling out.1092

A complex fraction is a rational expression whose numerator or denominator contains a rational expression.1103

Let's step back again and look at complex fractions when we are just talking about numbers, not yet working with polynomials.1110

An example of a complex fraction, just with numbers, would be something like 3/7, all that divided by 5/2.1117

Here I have a fraction, and the numerator of that fraction, and the denominator, are fractions; it is a complex expression.1126

I could also have something like 1/2 divided by 4; and here, just the numerator is a fraction.1134

That is still a complex expression: if either the numerator, the denominator, or both involve fractions, you have a complex expression.1143

So, looking at how to work with complex fractions that are algebraic complex fractions1152

(meaning that they involve rational expressions), let's look at this example.1160

(2x + 6)/(x² - 8x + 16), divided by (x² - 9)/(x² - 2x - 8):1166

now, this fraction bar, as you know, means "divided."1186

So, I have a complex fraction where both the numerator and the denominator are comprised of rational expressions.1190

To simplify a complex fraction, write it as a division expression and simplify.1199

So, I am looking at the main fraction bar and realizing that all this is saying is to divide.1204

So, I am going to rewrite this as (2x + 6), divided by (x² - 8x + 16),1210

all of this--this numerator--being divided by the denominator.1223

At this point, I am not doing anything except writing it out in a form that is more recognizable and easier to work with.1229

Once I have done this, this just becomes division with rational expressions.1239

And I know that, in order to divide one rational expression by another, I am going to multiply the first one by the inverse of the second...1243

times x² - 8x + 16...I am taking...this right here is x²...1256

the inverse of the second, minus 2x, minus 8, divided by x² - 9.1275

OK, so again, what I have done is taken the first rational expression, and I multiplied it by the inverse of the second, (x² - 2x - 8)/(x² - 9).1288

This is now just a multiplication problem; and recall that the next step is going to be to go ahead and factor, simplify, and multiply.1305

I can factor this out to 2 times (x + 3), divided by...well, this is going to be x, and this is x.1318

I have a positive sign here and a negative here, and what that is telling me is that I have a negative and a negative,1329

because when I multiply those two out, I am going to get a positive here; but when I add them, I will end up with a negative.1336

And this is actually (x - 4) (x - 4); I recognize that as (x - 4)².1343

Here, I have x times x, and I have a negative here, which tells me I am going to have a positive sign here and a negative here.1354

The factors of 8 are 1 and 8, and 2 and 4.1366

And I am looking for factors of 8 that add up to -2.1371

So, I am going to look at these two, because they are close together.1375

And I know that, if I take -4 and positive 2, and I add those up, I am going to get -2.1378

So, I am going to get x² - 4x + 2x, and -4x + 2x is going to give me -2x, and then -8 over here.1394

Down here, you probably recognize that this is (x - 3) (x + 3).1406

All right, so now I am going to look for common factors.1415

And the common factors that I have are going to be (x + 3) (I have an x + 3 here--I am going to cancel those out);1419

over here, I have (x - 4), and here I have an (x - 4), so those are going to cancel out.1438

So, I am looking around--are there any other common factors?--and there are not.1456

So, I am going to just multiply the numerator and the denominator: that is 2 times (x + 2), divided by (x - 4) times (x - 3).1459

Double-check at the end: do I have any common factors?1477

I do not, so this is in simplest form.1479

So again, I started out with a complex fraction; my first step was to simply rewrite that as a division problem.1482

So, I had the numerator right here, divided by the denominator, which is right here.1489

Then, I changed that to a multiplication problem; and with a multiplication problem,1498

I am going to set it up as the first rational expression (unchanged), times the inverse of the second rational expression.1506

Therefore, x² - 2x - 8 becomes the numerator, and x² - 9 becomes the denominator.1516

Once I have that, the next thing is to treat it like a regular multiplication problem, which is what it became.1526

So, I factored this out to (x + 3) times 2, factored the denominator, and did the same thing for the second one;1531

and then I looked through and saw that I had several common factors, so I went ahead and canceled those out.1541

OK, so that was handling complex fractions; and it just involved putting together what we have learned about division and multiplication of rational expressions.1548

Let's look at our first example: the first example just asks me to simplify a rational expression.1557

And the first step is going to be factoring.1563

I have 16 - x²: this is going to give me (4 + x) (4 - x).1567

And we are used to working with the opposite situation, x² - 16, but it is a very similar idea.1576

You can check that this is 4 times 4 (is 16), and then I multiply; that is -4x + 4x (those drop out), and then I get x times -x to give me -x².1583

So, that is factored correctly.1596

In the denominator, I have an x here, times an x here.1598

I have a negative sign in front of the constant, so I am going to have a plus here and a minus here.1603

Now, I need to just think about factors of 12.1609

And I need to look for factors of 12 that add up to -1.1614

And -1 is small, so I am going to look for factors that are close together; and that would be 3 and 4.1620

I am going to make the larger one negative, and the 3 positive, because I want to end up with a negative; and these equal -1.1626

I know that my correct factorization of the denominator would be (x + 3) (x - 4).1634

Now, to compare what is going on more easily, I am going to rewrite the numerator, putting the x's first.1643

I have (x + 4); here I am going to have (-x + 4).1650

This allows me to compare the factors, now that everything is in the same order--the terms within the factors are in the same order.1656

I can see here that I obviously don't have a common factor with this, and I don't have one with this.1666

But if I look at these, they are pretty close.1670

I again have the situation we discussed, where the signs are opposite.1673

I have a negative here; I have a positive here; I have a positive here; I have a negative here.1680

And remember: the way we handle that is to factor a -1 out from either this one or this one--you can choose either one.1689

I am going to go ahead and factor a -1 from the numerator--factor the -1 out from right here.1697

So, I am going to pull this out, and this is going to give me x - 4.1710

I now see that I have a common factor; so I can simplify by canceling out that common factor, leaving behind (x + 4) times -1...1716

I will pull that out in front...divided by (x + 3).1735

And just to write it even in a more standard form, instead of putting -1, I will pull the negative sign out in front.1740

And this becomes -(x + 4) divided by (x + 3).1746

OK, the first step is always factoring; then canceling out common factors--if you don't have a common factor,1753

see if factoring out a -1 from the numerator or denominator would give you a common factor (which it did).1760

And then, you can cancel those out.1767

And then, double-check at the end: I have no common factors, so I am done.1769

Example 2: we are asked to simplify here, and it is multiplication.1775

The first step is to factor; so in the numerator of the first rational expression,1780

I have a negative sign here; so I am going to put a + here and a - here.1788

Factors of 15 are 1 and 15, 3 and 5; I want those to add up to 2 (for 2x), so I know that these are too far apart; I am going to focus on this.1795

I want a positive sign here, so I am going to make the larger number positive to get 5 - 3 = 2.1810

So, I know I want the 5 here and the 3 here.1818

In the denominator, I have, again, the negative sign here; so it is (x + something) (x - something).1826

Factors of 10 are 1 and 10, 2 and 5.1835

I want them to add up to 3x, and so, again, I am going to look for the factors that are close together, which are going to be 2 and 5.1839

And I want a positive sign, so I am going to make 5 positive.1847

And those do add up to 3, so this is going to give me (x + 5) (x - 2) times...this time, I have a common factor of 4.1852

In the denominator, I have a common factor of 3.1863

Now, I have factored everything out; it is time to just cancel out common factors before I multiply.1867

(x + 5) and (x + 5) are common factors; (x - 3) and (x - 3) are also common factors.1875

(x - 2) and (x - 2)...so there are a lot of common factors.1884

And all I have left is 4/3; so what looked like a very complicated expression, actually, is just equivalent to 4/3.1888

Here we are asked to simplify, and this involves division.1900

So, when I see division, the first thing I do is rewrite it as multiplication: the first rational expression times the inverse of the second.1903

This becomes the numerator: x² + 12x + 36, divided by 3x² + 7x - 6.1915

OK, start factoring: and this factoring is a little more difficult, because the leading coefficient is not 1--it is 2.1926

I know I am going to have 2x here and an x here; I also know that I have a negative here.1936

So, one is going to be positive; one is going to be negative; but I don't actually know if the positive goes with the 2x or with the x.1942

I am going to have to do some trial and error up here.1948

I am trying to factor out 2x² + 5x - 3; and I know that I want the middle term to add up to 5x.1950

Factors of 3--there are not many, so this makes it a lot simpler--it is just 1 and 3.1959

So, trial and error: let's first put the 2x with the positive sign, and let's try the x with the negative sign.1965

I am going to go ahead and try 1 here and 3 here; and this is going to give me -6x + x, so that is going to add up to -5x.1976

I know that that is not correct, because the sign is wrong.1986

But I have the right number, 5; it is just that the sign is wrong.1989

So, I am going to try this again, but putting the negative sign with the 2x factor.1992

I am going to try this again: the outer term gives me 6x; the inner term is -x; and that equals 5x.2000

So, I know that this is the correct factorization.2007

The bottom is much simpler to factor: you will recognize that as (x - 6) (x + 6).2016

And when you multiply this out, the middle term would drop out, and you will just end up with x² - 36.2023

Here, I have x, and an x, and I have a positive sign here and here; so I am going to end up with a + and a +.2030

Factors of 36 (and I want these to add up to 12): I have 1 and 36, 9 and 4, and 6 and 6.2042

I know that these two add up to 37, so that is not correct; this adds up to 13; 6 + 6 is 12, so these are the correct terms within the factor.2058

OK, now this one, again, is a little bit more complicated, because the leading coefficient is not 1.2074

So, I have 3x and x, and I am dealing with a negative sign, so I have some kind of combination of positive and negative.2081

And I have 3x here, and I have x here; and I am thinking about the factors of 6 that are going to add up to positive 7.2096

So, I have 1 and 6, and 2 and 3; since I have a 3 that is going to amplify what I multiply, I am going to stay away from this 6 for right now.2122

I am going to work with 2 and 3; so let's try (3x + 2) and (x - 3).2135

This is going to give me...the first terms are 3x²; outer terms are going to be -9x.2142

The inner terms are + 2x, so that is going to add up to -7x.2150

What this is telling me is that I have the right combination, but the wrong signs.2156

So, I am going to try this again: (3x - 2) (x + 3).2160

Now, I multiply this out: 3x²...the outer terms give me 9x; the inner terms, -2x; and this gives me 7x.2166

So, this is my correct factorization; so you see, at this point in the course, we are using factoring as a tool to solve problems.2177

So, you really need to have the factoring down; and you can go ahead and review that in the Algebra I videos, if you need to,2185

especially for the more complicated problems, when the leading coefficient is something other than 1.2191

OK, so I have the factorization here: (3x - 2) and (x + 3).2198

So, I have this all factored out; now is the easy part--I just get to cancel out like factors.2205

I have (2x - 1); there is no common factor; I have (x + 3)--that is a common factor; I am going to cross that out and cancel it out.2211

I have (x + 6); there is a common factor--get rid of that; and one more (x + 6)--no common factor.2222

(x - 6) is no common factor; (3x - 2)--that is it; I am done with my common factors.2232

I multiply what is left over, which is (2x - 1) and (x + 6), divided by (x - 6) (3x - 2).2238

OK, so this took longer; it was more complicated; but it is really the same technique.2249

Start out by rewriting division as multiplication of the first rational expression, times the reciprocal of the second.2254

Factor; cancel common factors; and then, at the end, check and make sure you haven't missed any common factors.2261

Here we have a complex fraction involving rational expressions in both the numerator and the denominator.2271

The first step is always to rewrite this as a division problem.2277

Remember that the fraction bar is telling me to divide; so it is the entire numerator, divided by the denominator.2286

Recall that, in order to divide, we take the first rational expression (the first fraction) and multiply it by the inverse of the second.2300

At this point, we are just working with a multiplication problem.2313

The next step is to factor, and then simplify.2317

So, for the numerator here, there is no way to factor that out; let's go to the denominator.2321

I have a negative here, so this becomes a plus, and this is a minus.2331

And factors of 6 are 1 and 6, 2 and 3; I want factors that add up to -1.2336

These two are close together; so if I take -3 + 2, that equals -1.2345

So, my 3 is going to go by the negative; my 2 is going to go by the positive.2352

Now, to factor x² + x - 12...again, I have a negative sign, so I am going to do plus and minus.2359

Factors of 12 are 1 and 12, 2 and 6, 3 and 4.2369

And since I want these to add up to 1, I am going to pick the two that are close together, which are 3 and 4.2375

And I want this to be positive, so I am going to make the 4 positive and the 3 negative.2381

4 - 3 is 1, so it is (x + 4) (x - 3).2386

In the denominator, I have a greatest common factor of 4, so this becomes 4(2x - 3).2396

So, I did my factoring; and I see here that I have (2x - 3) in both the numerator and the denominator; those cancel.2405

I don't have any common factor with (x + 4); I do have a common factor with (x - 3); and this has no common factor.2413

Now, I multiply what is left behind: (x + 4) divided by...I have a 4 here and an (x + 2) there.2424

Finally, I double-check to make sure that there are no common factors that were missed.2433

I cannot simplify any more, so this complex fraction is now in simplified form.2437

That concludes this lesson on rational expressions, on multiplying and dividing.2447

And I will see you next lesson on Educator.com!2451