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### Multiplying and Dividing Rational Expressions

- To simplify an algebraic fraction, factor the numerator and denominator completely. Then cancel common factors. When multiplying or dividing, do the same thing – factor all the numerators and denominators, then cancel common factors.
- If two factors in the numerator and denominator look almost the same, factor –1 out of either of the factors and see if you get two identical factors.

### Multiplying and Dividing Rational Expressions

^{2}− 10n + 21)/(n − 3)]

- Factor the numerator
- [(n
^{2}− 10n + 21)/(n − 3)] = [((n − )(n − ))/(n − 3)] - [(n
^{2}− 10n + 21)/(n − 3)] = [((n − 3)(n − 7))/(n − 3)] - Cancel out common factors
- [((n − 3)(n − 7))/(n − 3)] = [((n − 7))/]

^{2}− 9)]

- Factor the denominator
- [(a + 3)/(a
^{2}− 9)] = [(a + 3)/((a + )(a − ))] - [(a + 3)/(a
^{2}− 9)] = [(a + 3)/((a + 3)(a − 3))] - Cancel out common factors
- [(a + 3)/((a + 3)(a − 3))] = [/((a − 3))]

^{2}− 25p)] ×[(p

^{2}− 7p + 10)/(5p + 25)]

- Factor the numerator and denominator completely
- [(p + 5)/(5p
^{2}− 25p)] ×[(p^{2}− 7p + 10)/(5p + 25)] = [(p + 5)/(5p( − ))] ×[((p − )(p − ))/(5(p + ))] - [(p + 5)/(5p
^{2}− 25p)] ×[(p^{2}− 7p + 10)/(5p + 25)] = [(p + 5)/(5p(p − 5))] ×[((p − 2)(p − 5))/(5(p + 5))] - Multiply
- [(p + 5)/(5p(p − 5))] ×[((p − 2)(p − 5))/(5(p + 5))] = [((p + 5)(p − 2)(p − 5))/(5p(p − 5)5(p + 5))]
- Cancel out common factors
- [((p + 5)(p − 2)(p − 5))/(5p(p − 5)5(p + 5))] = [((p − 2))/5p5]
- Simplify

^{3}− 2x

^{2})/(2x

^{3}+ 8x

^{2})]

- Factor the numerator and denominator completely
- [(3x + 12)/(3x + 9)] ×[(2x
^{3}− 2x^{2})/(2x^{3}+ 8x^{2})] = [(3( + ))/(3( + ))] ×[(2x^{2}( − ))/(2x^{2}( + ))] - [(3x + 12)/(3x + 9)] ×[(2x
^{3}− 2x^{2})/(2x^{3}+ 8x^{2})] = [(3(x + 4))/(3(x + 3))] ×[(2x^{2}(x − 1))/(2x^{2}(x + 4))] - Multiply
- [(3(x + 4))/(3(x + 3))] ×[(2x
^{2}(x − 1))/(2x^{2}(x + 4))] = [(3(x + 4)2x^{2}(x − 1))/(3(x + 3)2x^{2}(x + 4))] - Cancel out common factors
- [(3(x + 4)2x
^{2}(x − 1))/(3(x + 3)2x^{2}(x + 4))] = [((x − 1))/((x + 3))] - Simplify

^{2}− 6x + 8)/(x

^{2}− 16)] ×[(x + 4)/(5x

^{2}+ 25x)]

- Factor the numerator and denominator completely
- [(x
^{2}− 6x + 8)/(x^{2}− 16)] ×[(x + 4)/(5x^{2}+ 25x)] = [((x − )(x − ))/(( + )( − ))] ×[(x + 4)/(5x( + ))] - [(x
^{2}− 6x + 8)/(x^{2}− 16)] ×[(x + 4)/(5x^{2}+ 25x)] = [((x − 4)(x − 2))/((x + 4 )(x − 4))] ×[(x + 4)/(5x(x + 5))] - Multiply
- [((x − 4)(x − 2)( x + 4 ))/((x + 4 )(x − 4)5x(x + 5))]
- Cancel out common factors
- [((x − 4)(x − 2)( x + 4 ))/((x + 4 )(x − 4)5x(x + 5))] = [((x − 2))/(5x(x + 5))]
- Simplify

^{2}− 1)/(x

^{2}+ 2x + 1)] ×[(x

^{2}+ 2x + 1)/(x + 1)]

- Factor the numerator and denominator completely
- [(x
^{2}− 1)/(x^{2}+ 2x + 1)] ×[(x^{2}+ 2x + 1)/(x + 1)] = [((x + )(x − ))/(( + )( + ))] ×[(( + )( + ))/(x + 1)] - [(x
^{2}− 1)/(x^{2}+ 2x + 1)] ×[(x^{2}+ 2x + 1)/(x + 1)] = [((x + 1)(x − 1))/((x + 1)(x + 1))] ×[((x + 1)(x + 1))/(x + 1)] - Multiply
- [((x + 1)(x − 1))/((x + 1)(x + 1))] ×[((x + 1)(x + 1))/(x + 1)] = [((x + 1)(x − 1)(x + 1)(x + 1))/((x + 1)(x + 1)( x + 1 ))]
- Cancel out common factors
- [((x + 1)(x − 1)(x + 1)(x + 1))/((x + 1)(x + 1)( x + 1 ))] = [((x − 1))/]
- Simplify

^{2}− 9x)/(x

^{2}− 7x + 12)] ÷[(5x

^{2}− 20x)/(x − 4)]

- Factor the numerator and denominator completely
- [(3x
^{2}− 9x)/(x^{2}− 7x + 12)] ÷[(5x^{2}− 20x)/(x − 4)] = [(3x( − ))/(( − )( − ))] ÷[(5x( − ))/(x − 4)] - [(3x
^{2}− 9x)/(x^{2}− 7x + 12)] ÷[(5x^{2}− 20x)/(x − 4)] = [(3x(x − 3))/((x − 3)(x − 4))] ÷[(5x(x − 4))/(x − 4)] - Get rid of the division by multiplying by the recriprocal of the second rational
- [(3x(x − 3))/((x − 3)(x − 4))] ÷[(5x(x − 4))/(x − 4)] = [(3x(x − 3))/((x − 3)(x − 4))] •[((x − 4))/(5x(x − 4))] = [(3x(x − 3)(x − 4))/((x − 3)(x − 4)5x(x − 4))]
- Cancel out common factors
- [(3x(x − 3)(x − 4))/((x − 3)(x − 4)5x(x − 4))] = [3/((x − 4)5)]
- Simplify

^{2}+ x − 12)/(x − 3)] ÷[(x

^{2}+ 6x + 8)/(x

^{2}+ 4x + 4)]

- Factor the numerator and denominator completely
- [(x
^{2}+ x − 12)/(x − 3)] ÷[(x^{2}+ 6x + 8)/(x^{2}+ 4x + 4)] = [(( + )( − ))/(x − 3)] ÷[(( + )( + ))/(( + )( + ))] - [(x
^{2}+ x − 12)/(x − 3)] ÷[(x^{2}+ 6x + 8)/(x^{2}+ 4x + 4)] = [((x + 4)(x − 3))/(x − 3)] ÷[((x + 2)(x + 4))/((x + 2)(x + 2))] - Get rid of the division by multiplying by the recriprocal of the second rational
- [((x + 4)(x − 3))/(x − 3)] ÷[((x + 2)(x + 4))/((x + 2)(x + 2))] = [((x + 4)(x − 3))/(x − 3)] = [((x + 4)(x − 3)(x + 2)(x + 2))/(( x − 3 )(x + 2)(x + 4))]
- Cancel out common factors
- [((x + 4)(x − 3)(x + 2)(x + 2))/(( x − 3 )(x + 2)(x + 4))] = [((x + 2))/]
- Simplify

- Rewrite the complex fraction by dividing the numerator by the numerator using the division sign ÷
- [([3/(3x + 15)])/([(3x + 6)/(2x + 10)])] = [3/(3x + 15)] ÷[(3x + 6)/(2x + 10)]
- Factor the numerator and denominator completely
- [3/(3x + 15)] ÷[(3x + 6)/(2x + 10)] = [3/(3( + ))] ÷[(3( + ))/(2( + ))]
- [3/(3x + 15)] ÷[(3x + 6)/(2x + 10)] = [3/(3(x + 5))] ÷[(3(x + 2))/(2(x + 5))]
- Get rid of the division by multiplying by the recriprocal of the second rational
- [3/(3(x + 5))] ÷[(3(x + 2))/(2(x + 5))] = [3/(3(x + 5))] •[(2(x + 5))/(3(x + 2))] = [(3*2(x + 5))/(3(x + 5)*3(x + 2))]
- Cancel out common factors
- [(3*2(x + 5))/(3(x + 5)*3(x + 2))] = [*2/(*3(x + 2))]
- Simplify

^{2}+ 20x)])/([(5x − 15)/(25x + 20)])]

- Rewrite the complex fraction by dividing the numerator by the numerator using the division sign ÷
- [([5x/(25x
^{2}+ 20x)])/([(5x − 15)/(25x + 20)])] = [5x/(25x^{2}+ 20x)] ÷[(5x − 15)/(25x + 20)] - Factor the numerator and denominator completely
- [5x/(25x
^{2}+ 20x)] ÷[(5x − 15)/(25x + 20)] = [5x/(5x( + ))] ÷[(5( − ))/(5( + ))] - [5x/(25x
^{2}+ 20x)] ÷[(5x − 15)/(25x + 20)] = [5x/(5x(5x + 4))] ÷[(5(x − 3))/(5(5x + 4))] - Get rid of the division by multiplying by the recriprocal of the second rational
- [5x/(5x(5x + 4))] ÷[(5(x − 3))/(5(5x + 4))] = [5x/(5x(5x + 4))] •[(5(5x + 4))/(5(x − 3))] = [(5x*5(5x + 4))/(5x*(5x + 4)5(x − 3))]
- Cancel out common factors
- [(5x*5(5x + 4))/(5x*(5x + 4)5(x − 3))] = [*/(*(x − 3))]
- Simplify

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Multiplying and Dividing Rational Expressions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Simplifying Rational Expressions
- Factoring -1
- Multiplying and Dividing Rational Expressions
- Multiplying and Dividing
- Example: Multiplying Rational Expressions
- Example: Dividing Rational Expressions
- Factoring
- Complex Fractions
- Example 1: Simplify Rational Expression
- Example 2: Simplify Rational Expression
- Example 3: Simplify Rational Expression
- Example 4: Simplify Rational Expression

- Intro 0:00
- Simplifying Rational Expressions 0:22
- Algebraic Fraction
- Examples: Rational Expressions
- Example: GCF
- Example: Simplify Rational Expression
- Factoring -1 4:04
- Example: Simplify with -1
- Multiplying and Dividing Rational Expressions 6:59
- Multiplying and Dividing
- Example: Multiplying Rational Expressions
- Example: Dividing Rational Expressions
- Factoring 14:01
- Factoring Polynomials
- Example: Factoring
- Complex Fractions 18:22
- Example: Numbers
- Example: Algebraic Complex Fractions
- Example 1: Simplify Rational Expression 25:56
- Example 2: Simplify Rational Expression 29:34
- Example 3: Simplify Rational Expression 31:39
- Example 4: Simplify Rational Expression 37:50

### Algebra 2

### Transcription: Multiplying and Dividing Rational Expressions

*Welcome to Educator.com.*0000

*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 (2x ^{2} - x + 9), over (x^{2} - 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 that*0161

*x ^{2} - 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 here*0321

*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: 4xy ^{3} times x^{2}.*0514

*And that is going to give me...times 12xy ^{5}, divided by x^{4}.*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 x ^{2} times x^{4}, 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 x ^{2} 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 4y ^{3} times 12y^{5}, divided by x^{4}.*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 base*0600

*means that I am going to add the exponents, so this is going to give me y ^{8} divided by x^{4}.*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 2xy ^{2} divided by 4x^{2}y^{3} times xy divided by 8x^{2},*0680

*divided by...if I am being asked to divide these, I am simply going to rewrite this*0703

*as 2xy ^{2} divided by 4x^{2}y^{3}; so I keep that first rational expression the same.*0708

*And then, I multiply it by its inverse, which is 8x ^{2} 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 x ^{2} 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 y ^{3}.*0785

*And actually, I have one more common factor left: so 4y/y ^{3}.*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 y ^{2}.*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 ahead*0850

*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, 2x ^{2} - 6x - 20, divided by x^{2} - 49:*0876

*if I am asked to multiply that by, say, 5x - 35, divided by x ^{2} + 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 x ^{2} - 3x - 10.*0910

*x ^{2} - 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 x ^{2} + 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 fractions*1152

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

*(2x + 6)/(x ^{2} - 8x + 16), divided by (x^{2} - 9)/(x^{2} - 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 (x ^{2} - 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 x ^{2} - 8x + 16...I am taking...this right here is x^{2}...*1256

*the inverse of the second, minus 2x, minus 8, divided by x ^{2} - 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, (x ^{2} - 2x - 8)/(x^{2} - 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 x ^{2} - 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, x ^{2} - 2x - 8 becomes the numerator, and x^{2} - 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 - x ^{2}: this is going to give me (4 + x) (4 - x).*1567

*And we are used to working with the opposite situation, x ^{2} - 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 -x ^{2}.*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: x ^{2} + 12x + 36, divided by 3x^{2} + 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 2x ^{2} + 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 x ^{2} - 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 3x ^{2}; 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: 3x ^{2}...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 x ^{2} + 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

1 answer

Last reply by: Dr Carleen Eaton

Sun Jul 8, 2018 12:59 PM

Post by John Stedge on June 12 at 02:00:12 PM

In example one can't you simplify further and make the answer -4/3 instead of -x+4/x+3?

1 answer

Last reply by: Dr Carleen Eaton

Sun Feb 11, 2018 9:50 PM

Post by Shiden Yemane on February 10 at 12:32:39 PM

Thanks Dr. Eaton! I've learned a lot from your Algebra 2 series.

0 answers

Post by julius mogyorossy on December 25, 2014

Merry Christmas Dr. Carleen, thanks for everything, I can't wait till you know who I am, how awesome I am. I got my math notes on my iPod, now if I can only find where apple is hiding them from me, apple is always hiding things from me. I can't wait to take the Clep test. I wish I could hire you as my coach.

1 answer

Last reply by: Dr Carleen Eaton

Thu Mar 27, 2014 6:46 PM

Post by Taslim Yakub on February 21, 2014

why does the 4+x become x+4 and not have a negative sign in front of it.

1 answer

Last reply by: Dr Carleen Eaton

Wed Jan 1, 2014 1:00 AM

Post by Myriam Bouhenguel on December 27, 2013

for example 3 it is supposed to be (2x+1) (x-3) for the factorization of 2x^2+5x-3 at 34:37