In the previous lessons we worked with polynomial expressions. The variables had non negative integer exponents. And our expressions could contain any number of terms. We learned how to add, subtract, multiply and even divide these polynomials. In the upcoming lessons, we'll make use of our new factoring and look at performing these same operations for rational expressions. A rational expression is a ratio of two polynomials. It is a fraction in which the numerator and the denominator are both polynomials, 1 over x is an example of a rational expression, since 1 is a mynomial and x is also a mynomial. The second example contains the polynomial x plus 2 in the numerator, and 3x minus 1 in the denominator. So this is also a rational expression. And this course will prepare you to handle any sort of polynomial or rational expression. You'll be able to add, subtract, multiply and even divide these rational expressions by the end of this unit.
Before we go into more depth about rational expressions, let's recall our knowledge about fractions. A fraction is a rational number or a ratio of two integers. In this case, we have the ratio of five to eight or five eighths. A student showed the following work when trying to simplify five eighths. Do you think this work is correct, and why? Choose yes or no and then check off either reasons that support your answer.
This work would not be correct, since we know that four-eighths would be equal to one-half. More importantly, we know that this step cannot be correct. The student tried to simplify terms in the numerator and in the denominator. Remember terms are separated by addition or subtraction. We cannot cancel terms, but we can cancel factors. Factors are numbers or variables that are multiplied in the numerator or denominator. For example, if we started with six-12ths. We can rewrite the 6 as 2 times 3, and 12 as 2 times 6. We can simplify the factors of 2, since these cancel out to equal 1. This would leave us with 3 6. We repeat this process one more time by rewriting 3 and 6 with factors. We can cancel the whenever we're working throughout this unit keep in mind that we can't cancel terms, we can only cancel factors, things that are multiplied in the numerator and in the denominator. And finally, we know 5/8 cannot be simplified since 5 and 8 do not share a common factor. So we also need to include this reason.
We can use our knowledge of canceling factors to help us simplify rational expressions. We simplify a rational expression the same way that we simply fractions. We just need to cancel the factors that appear in the numerator and the denominator since the factors reduced to 1. We can use these steps for any rational expression we encounter. So let's start off with this one. I want you to factor the numerator and the denominator in the set here.
Well we know we can factor a 3 from both the 1st term and the 2nd term leaving us with x minus 4 for our 2nd factor. For the denominator we know that they share a common factor of 5 so we put that out in front which leaves us with x minus 4 for our 2nd factor. So, this would be our factor rational expression great job if you found it.
Now that we have factored each part of our fraction, we can remove the common factors or simplify them since they equal one. For example, we know that this quantity that X minus four divided by the quantity X minus four would equal one. This would leave us with three times one which is three and five times one which is five, or three 5th. We know we can do this because X really represents a number, so if X were five we would be left with three times one and five times one, which is 3 5th. But x doesn't have to be 5, it could be any number. If we let x equal 6 then we plug in 6 for x and we get 3 time 6 minus 4 divided by 5 times 6 minus 4. This is really the same as 3 times 2 divided by 5 times 2. The twos are factors that reduce to 1, so we really have 3 fifths. It wouldn't matter what the value of x was. The quantity x minus 4 divided by the quantity x minus 4 would always equal 1.
But this simplification to 3 5th wont be true for every value of X. There is actually one value of X that makes it not true. What value do you think it is? You'll want to think back to what you know about fractions and what you know about their denominators. Take some time to think, and when you're ready, put your answer here.
We know that the denominator of a fraction cannot equal 0. So, if we solve this for x, we know x cannot take on the value of 4. Now, I didn't show the steps of adding 20 to both sides and then dividing by 5, but I think that you can easily see that if x were 4, we would get 0, which can't happen for the denominator. This idea of excluded values of x is really important. Later, when we solve equations with rational expressions, we want to make sure we don't allow certain values for the variable or for x. We don't want the denominators to be 0.
Let's try simplifying this rational expression. First, I just want you to factor the numerator and the denominator. Write your factors here.
To factor the first trinomial, we want to find factors of 30 that sum to negative 11. These factors are negative 5 and negative 6, and since our quadratic term has a 1 in front of it, we can just use these factors in our factored form here. X minus 5 times x minus 6. For the denominator we want to find factors of 1 times 20, or 20. That sum to negative 9. Those would be negative 4 and negative 5. And again since our quadratic term has a 1 as the coefficient, we'll use these factors in our factored form. Great job factoring if you got one or both of those correct.
Now, that our fraction has been factored, we can cancel the factors that appear in the numerator and in the denominator. You can write your answers here.
We can see that there's a common factor of x minus 5 in the numerator and in the denominator. So we know those simplify to 1. This would leave us with x minus 6 in the numerator and x minus 4 in the denominator. Great job on that simplification.
Now, that we have this fraction, can we make this simplification to get 3 halves, yes or no?
Well that answer would be no. We can't do this simplification. We cancelled terms here. Remember we can only cancel factors, or things that are multiplied in the numerator and in the denominator. X minus 6 divided by the quantity x minus 4, would be our final answer. If you're ever trying to simplify a fraction further, ask yourself if you can factor the numerator and the denominator, or if you can cancel factors that appear in both. In this case, we can't factor anything besides 1 out of the numerator and the denominator, and these 2 factors aren't the same. So, this is our final result.
Try simplifying this rational expression. Remember to only cancel factors that appear in the numerator and in the denominator
The first thing we want to do is factor our numerator and denominator. To factor the numerator, we find factors of negative 8 that sum to positive 2. And since the coefficient in front of m squared is 1, we can use these two factors in our binomials. For the denominator, we want to find factors of negative 12 that sum to negative 1. These factors are negative 4 and positive 3. Now, we can't just use these factors in our binomials since the m squared has a coefficient of 2. Instead, we'll need to use factoring by grouping. Using that method, we get a denominator of 2 m plus 3 times m minus 2. Now that we have our factored form, we can cancel factors that appear in the numerator and denominator. Since they equal 1. And this leaves us with our solution. We know we can't simplify the m's anymore since we have a sum and the numerator and the denominator. We don't have anymore factors to cancel.
In the last few examples, we saw when rational expressions had a factor in the numerator and the denominator that equaled 1. This is true for the quantity m minus 2 divided by m minus 2. We know this fraction really equals 1. But what do you think about for these expressions? What do you think these would equal? Write your answer in the boxes.
Well it turns out all of these are negative 1. For the first fraction we can think about simplifying the common factor of 6. Leaving us with negative 1 divided by 1 or just negative 1. We can use the same reasoning for the rest of the examples. They will all equal negative 1 since we'll have one factor that's positive in the numerator. And one factor that's negative in the denominator. It doesn't matter where the negative sign falls, we just need one of them to be negative in order for our answer to be negative one.
Instead of just looking at numbers, now, let's look at polynomials. What do you think these expressions would equal? Write your answers here.
The second one might have been tricky but they both equal negative 1. We want to remember that the negative sign is really a negative 1 being multiplied by this expression. We know x plus 3 divided by x plus 3 equals 1. So now we can simplify our fraction. This just leaves us with negative 1. For our second rational expression, we can think of a 1 being outside of the expression a plus negative 1 a plus 4. If we were to distribute this negative 1 back inside the parenthesis we'd get our original denominator. We know this step is correct. We can simplify the a plus 4s since they equal 1leaving us with 1 times 1 in the numerator and negative 1 times 1 in the denominator. This just simplifies to negative 1. You might have spotted this first one easily. Great work if you. If you did. It's okay if this one gave you trouble.
We're going to focus in on this idea of opposite polynomials. For two polynomial expressions to be opposite, every corresponding pair of terms must opposite in sign. We know addition is commutative so we can rewrite these terms by adding the opposite. I'll move the negative x squared to the front, the 3x to the middle, and the negative 8 to the end. Notice that when we move these terms around, we just keep the sign with it. The three x's are positive, the eight is still negative and the x squared is also still negative. Now we can definitely see that these two polynomials are opposites. The x squared terms are opposites since one is positive and the other is negative. The three x's are opposites since one is negative and the other is positive And the eights are opposites, since the first one's positive and the second one's negative. And the neat part about opposite polynomials and rational expressions is that they simplify to negative1 . We can think about factoring a negative 1 from each of these terms. So we'll have negative 1. Times the opposite sign of all those terms in the denominator. The numerator will just stay the same and we can think about multiplying it by a positive 1. Here we have a common factor in the numerator and a denominator, so this all equals 1. And finally, since we have this negative sign down in the denominator, our answer simplifies to negative 1.
For two polynomial expressions to be opposites every corresponding pair of terms must be opposite in sign. Knowing this, I want you to determine whether each of these rational expressions is equal to positive one, negative one or something else. Choose one of these for each expression. Good luck.
For the first expression, we can switch the terms in the denominator, since addition is commutative. So we'll have x plus 7 divided by the quantity x plus the second rational expression, we know it equals negative 1. This b squared is positive and this b squared is negative, whereas this 5 is negative and this 5 is positive. So, we definitely have opposite polynomials. We see that here, if we just reverse the terms of the denominator and keep the signs in front of them. We factor a negative 1 from the denominator and then the b squared minus have thought it was negative 1. You might have cancelled the terms which are a and then simplified negative 5 over positive 5. But, remember, we can't cancel terms. We can only cancel factors. A is positive in the numerator and positive in the denominator, so, these aren't opposite polynomials. The 5s have different signs which mean this polynomial can't equal positive 1. It must be something else. And for the last one, we could switch the terms in the denominator, and then, factor a negative 1 from the numerator. The common factors of 2x minus 9 simplify to 1, leaving us with negative 1 as our result. Great work if you even got two of those correct. These were pretty tough.
What about these rational expressions? Are they equal to positive 1, negative 1, or something else?
The first two are equal to negative 1, the 3rd one is equal to positive 1, and the last one is something else. We know these first two are equal to negative 1 since each rational expression contains opposite polynomials. The x terms are opposite in sign here, and the y terms are also opposite in sign. So we know that this fraction can simplify to negative 1. The same is true for this fraction. The b squared terms are opposite in sign, the 5b terms are opposite in sign and the constant of 1 is also opposite in sign. I didn't show the factoring here, but we know that it will equal negative 1 if we remove a factor of negative 1 from the denominator and cancel our trinomial. In our third example, the y terms have the same sign, and the x terms have the same sign. So really, this fraction is the same expression. In this third rational expression, the y terms have the same sign, as do the x terms. If we switch the order of addition in the numerator, we'll have x minus y divided by the quantity x minus y. We know these expressions are exactly the same, so we really have 1 divided by 1, or just positive 1. For this last expression, we know it needs to equal something else. We can't just square each term and call it a day. We know that this is really x plus 2 times x plus 2. When we square a binomial, we get a perfect square trinomial. Looking at this numerator and denominator, we know that our fraction doesn't equal positive 1 or negative 1.
Let's continue our exploration of rational expressions. This time I'm going to show you 4 rational expressions and I want you to find the one expression that is not equal to the other 3. In other words you want to try and either to simplify these to be positive 1 or negative 1 or you want to rearrange them to make them look like each other. Be sure to chose the odd one out. The one that looks different.
This expression is the odd one out. The other ones equal positive 1. This expression equals 1 since we have the same expression in the numerator and the denominator. We've seen something similar like this before if we just switch the terms in the denominator we can see that numerator and the denominator have the same factors. Notice that I'm not cancelling an individual term. I'm cancelling an entire numerator with an entire denominator. So I have one divided by one or just one. This one might have been a little bit tricky. Here we have a negative one times our rational expression. Well we can just distribute this negative sign or this negative one to each term and the numerator. This would give us x plus 3 Divided by 3 plus x. This fraction is identical to this rational expression, so we know it equals 1. For this rational expression, we know that these factors can reduce to 1, but we're left with a negative sign out in front, or a negative 1 times 1, which is just negative 1.
What about for these four expressions? Which one of these is different?
We know this is the answer. This rational expression is the odd one out, since it equals positive 1. This expression over here is really equal to negative 1, since x minus 7 divided by the quantity x minus 7 simplifies to 1. We get a negative 1 since we have a negative sign sitting out here. This is really negative 1 times 1. For this first rational expression, we can switch the order of the denominator. I can have the negative x come first, and then followed by the positive 7. We can see that this expression is the same as this last expression down here. But this doesn't quite look like negative 1. But these two rational expressions don't look like negative 1, but in fact, they are. We know these expressions in the numerator and denominator are opposite polynomials. The x terms are opposite in sign, as are the constant terms. We can factor a negative 1 from our denominator to get negative 1 times x minus 7. Then we can simplify our common factors in the numerator and denominator, leaving us with negative 1. Nice thinking if you found that one.
What about for these four expressions? Which one is different? And as a hint, these won't simplify to positive 1 or negative 1. You'll need to manipulate each rational expression to see if they're the same.
We can see that these two rational expressions are equal. If we switch the order of the terms in the numerator of this expression, we'll get x plus 9 divided by the quantity x minus 9. Now we need to decide between these two expressions. We can switch the order of the terms in the numerator and in the denominator. The x stays negative here and the 9 stays positive. For this expression, we don't want to switch the order of the numerator, since it's already x plus 9. It matches the other ones. And here's where this negative sign comes in handy. We can distribute this negative sign to each term in the numerator, or to each term in the denominator. I'm going to multiply our denominator by negative 1. Negative 1 times 9 is negative 9. And negative 1 times negative x is positive x. Finally, we commute this edition to get x plus 9, divided by the quantity x minus 9. So we know this rational expression is equal to these two. This one is definitely different.
Now that we've seen different types of rational expressions that equal 1 and negative 1, let's try and define some others. What do you think this would equal? Write your numerator here and your denominator here. Good luck.
This first trinomial can be factored as x plus 4 times x plus 6. We want factors of positive 24 that sum to positive 10. The second trinomial can be factored into x plus 5 times x plus 4. We want factors of positive 20 that sum to positive 9. Once we have our factored form, we cancel the factors that appear in the numerator and the denominator so we're left with x plus 6. Divided by the quantity, x plus 5. Great work if you've got this as your final rational expression.
Try factoring this expression. Remember to use all your factoring skills and start by looking for a GCF. Good luck.
This is our simplified expression, 8a divided by the quantity a plus 5. We can start factoring the numerator by removing a common factor of 8a from each term. This leaves us with a second factor of a plus 4. This denominator is actually identical to the last problem we looked at. The x has been replaced with a. So we have the same factors, a plus 5 and a plus 4. We just have a instead of x. We cancel the common factor of a plus four in the numerator and denominator, leaving us with eight a divided by a plus five. Keep in mind these are both positive ones, so we wouldn't need to change anything in terms of signs. Sometimes our two factors might reduce to negative one. In that case we would need to multiply our expression by a negative one. We would have a negative sign in front.
Try simplifying this expression. Write your answer here.
Here's our answer. Great work if you got that one correct. We can factor the numerator to be x plus 3 times x minus 3, since this is the difference of two squares. Our denominator factors into x plus 3 squared. This is a perfect square trinomial. The first term is a square. The second term is a square, and the middle terms is twice the square root of the first term and the last term. We know for the square exponent, we really write this factor twice. From here, it's easy to see we have a common factor of x plus 3, in the numerator and the denominator. So those simplify to equal 1. Another way to think about simplifying this is to do it over here. We have an exponent of 1 for this x plus with the 1 and the numerator. So our power of 2 drops down to a power of 1 and we lose this factor altogether. Keep in mind that things aren't equal in 0 here, it's really that we have an equivalence of 1, x plus 3 divided by x plus 3. This leaves us with our final answer.
Try simplifying this expression. Write your answer here and be sure to factor out a greatest common factor first.
We can remove a 4 from the first 3 terms of out numerator, leaving us with 4 times x squared plus 4x plus 4. We can also remove a 4 as a greatest common factor from 4x and 8. This leaves us with 4 times x plus 2 in our denominator. We have a quadratic trinomial here and not only that it's a perfect square trinomial, so we can factor that as x plus 2 times x plus 2. Finally, we cancel the factors in the numerator and the denominator. So x plus two divided by x plus two equals one, and four divided by four equals one. Remember, we're only simplifying factors, or things that are being multiplied in our numerator and our denominator. This leaves us with x plus two divided by one, or just x plus two. Notice that not every rational expression simplifies to another fraction. Sometimes we'll just get another polynomial.
What about this expression? What do you think it equals?
We can factor this numerator as the difference of two squares. We can write it as 6 minus x times 6 plus x. For this denominator we want to find the factors of negative 48 that sum to positive 2. These factors are negative 6 and positive 8. When I look at my factors it doesn't seem as if anything can simplify out to equal 1 but if i look at these two factors i can see that these are opposite polynomials. The constant term 6 is positive while this one's negative and the x term is negative here while this x term is positive. This means I'll have a factor of negative 1. We can see this if we were to just switch the order of the terms for these two, we'll have negative 6 plus x, then I can factor a negative signs of both of these terms since i factored a negative 1 out. These factors of I thought here. Whenever you see opposite polynomial factors you can simplify them to be negative 1. We don't need to show all this work. So we're left with this expression. Now, I want to acknowledge there are a lot of different answers here. This negative sign could have been distributed to either the numerator or the denominator. If we distribute the negative 1 to the numerator, we could have gotten this expression. Or switch the order of these terms and you got this expression. These are all equal. If we distribute this negative one instead of the denominator, we can wind up with any of these expressions. And in fact, all of these are correct. In general, it's easiest to list the negative sign out in front of a fraction and then write your rational expressions.
What about this rational expression? What would you get? Notice we've changed one thing here, the sign. This is a negative 2x instead of a positive 2x.
Our solution is 6 minus x, divided by the quantity x minus 8. Great job if you got this rational expression. We factor the numerator the same as before, but this time in the denominator we want factors of -8 and positive 6. They multiply to give us -48, and a sum to get us -2. I noticed that x plus 6 and 6 plus x are really the same factors.So I can cancel those since they equal 1. This leaves me with the remaining factors, so I have 6 minus x divided by x minus 8. You might have switched the order of these two terms, giving you negative x plus 6 for the numerator. This would be okay. You might even have gone a step further, and factored negative 1 from the numerator, and that would have been okay too. All of these answers are correct. And if you even followed my advice from the first time, a negative sign out in front would look like this.
For our last problem, try simplifying this expression.
Our numerator is a difference of two perfect squares. So we can factor it as 2x plus 5y times 2x minus 5y. This denominator isn't identical to one of our factors, but it is an opposite polynomial for this factor. We can see that the factors simplify to be negative 1 and were left with negative 1. Times 2x plus Fantastic work if you got that expression.