Understanding Algebraic Expressions and Operations

So the topic is about algebraic expressions. So we'll see how different types of algebraic expressions are expressed and then how we can add them and subtract them. So before that we need to see some basic properties on algebraic expressions. So there are totally many different types of algebraic expressions. To start with the first algebraic expression is a monomial. A monomial is an algebraic expression which has only one term. So for example, algebraic expression with only one term. For example, if I just take 3x, 5x squared. and minus 7x or I take 11. So all these are monomials because they are just written with only one term. So same way if I just say binomial, it is an algebraic expression with two terms is called a binomial. So let's start with Monomial and the next is binomial and then comes trinomial and so on and so forth. So binomial is an algebraic expression with two terms. So this is the definition for a binomial expression. So for example, if I just take... 2x minus 3, I identify here that there are two terms, one is 2x and one is negative 3. So the combination of two terms written in an expression form is called a binomial expression. So an expression written with only one term is called a monomial expression. So I would like to take one more example where 5x squared minus 7 also is a binomial. If I have two unknown variables, say 2x plus 3x square is also a binomial, where the first term is 2x and the second term is 3x square. Let me take one interesting example where x square plus 3x minus 4x square. Now here, I identify that This is not in the binomial form, but I can make it converted into binomial expression. Because these two terms can be further simplified to reduce to one particular term, so that you get two terms reduced after simplification. So this indirectly is a binomial expression in the way that, when I just take x square minus 4x square is minus 3x square, and plus 3x written out here. So the simplification of x square and negative 4x square gives me minus 3x square plus 3x fixed and written as the second term, which makes me understand that this is the first term and the second term. Because it has two terms in the expression, this is an example of a binomial expression. Whereas if you take the unsimplified form, it doesn't form a binomial. expression. So, these are the various examples for binomial. Now, once monomial and binomial has been discussed, the third thing what is there in the definition is trinomial. And you all can guess what a trinomial is based on the definitions which we have taken there. An algebraic expression with one term is monomial, with two terms is binomial, so with three terms will be a trinomial. So the definition is quite easy. Algebraic expression with three terms and even there are so many examples which we can take out here. If I take an example where 5 minus 3x plus 7x squared is a trinomial because it has three terms as it can be clearly seen. The first term is 5, second term is negative 3x and the third term is 7x power 2. So these three terms makes this algebraic expression to be called as a trinomial or a trinomial expression. So there are many many examples which we can take in mathematics related to trinomial. Yes, so these are the three basic examples which are taken. One is the monomial, binomial and trinomial. So there are many expressions where you can have more than three, you can have four, five, six. So generally any expression which has more than three terms is called a multinomial. So we need to, we need not, we cannot take the number of terms defined for any number of expressions where the terms are increased to a larger extent. It is worth meaningful to take one term as monomial, two terms as binomial and three terms as trinomial. But when you have n number of terms, it is called a multinomial. So an algebraic expression with n terms with n greater than 3 is called a multinomial. Where for n equal to 1, 2, 3 is already defined for monomial, binomial and trinomial. So let's see an example for a multinomial. For example, I take an algebraic expression 3 minus 4x plus 5x squared plus 7x cubed minus 9x power 4 minus 9x power 4 plus 12x power 5. Now this algebraic expression, let's see how many terms are there and we'll count by the process. 1, 2, 3, 4, 5, 6. So this is an algebraic expression with six terms which is called a multinomial with six terms. So this is an example of a multinomial consisting of six terms. Now, based on all the various types of algebraic expressions which we have seen, starting with monomial, binomial, trinomial and multinomial, we will now see how we are going to add or subtract the different algebraic expressions using the like terms and unlike terms. So first, initially, we will see what exactly is a like term and what exactly is the unlike terms. So when we discuss about like terms, the terms where the degree is same are called the like terms. So if the degree is same, then the terms are like terms. So when I say degree, the highest power of the unknown variable, say for example, I take 3x minus 5x square, 4x, 7x cube. So in this case, I find that the degree of this term is 1 because the power of x. is 1 here. Now when I take here the degree of the second term is 2 because the power of the unknown variable that is x is 2. If I take here the degree of this term is 1 because x has its power as 1. When I take this term, its degree is 3 because of x cube. Now in this case, I identify that the first term and the third term are having the same degree. So these two are said to be the like terms. So the first term and the third term have same degree 1. Therefore, according to the definition of like terms, 3x and 4x are called Like terms. So whenever we have any problem where we need to identify the like terms, we just search for the degree and if the degree is same, they are like terms and if the degree is not same, they are not the like terms. So when we define the like terms, similar fashion we can define the unlike terms. If the degree is not same, then the terms are unlike terms. So here when I take 3x and minus 5x square these two are having two different degrees. Therefore these two are unlike terms. These two are like terms. 3x and 7x cube are unlike terms because the degree of this is 1 and degree of this is 3. So let's take it as a different example definition and then see many examples of unlike terms. So let's see the definition of unlike terms, the way we are seen for like terms. So unlike terms are the terms whose degrees are not same. So let's take the definition, the terms whose degree... are not same. They are different. They are called unlike terms. Say for example, 5x and 7x cube. I find that the degree of 5x is 1 and the degree of 7x cube is 3. So these two are with different degrees. So these two are called the unlike terms. So therefore they are different. unlike terms the reason being with two different degrees So after we understood the concept of like terms and unlike terms, the concept of like terms is very much useful when we add or subtract the two algebraic expressions or more than two algebraic expressions. Because when we add two algebraic expressions, we write all the like terms in column wise. And let's see how we are going to do that. So the topic here is adding of. Algebraic expressions. So before we proceed with how we are going to add, we are going to take certain rules into consideration. Step 1. We write all like terms together. Write all like terms. of each algebraic expression one below the other. So we write all like terms one below the other how many our algebraic expressions we have that is the first step we are going to follow. Step 2 After we write all the like terms one by one, one below the other, we add the coefficients of each term individually and then write the power of x besides. Add the respective coefficients of like terms and write the exponent whichever is the degree respectively so after that we obtain the sum of two or more than two algebraic expressions so these are the two important steps we need to follow before we proceed with adding so let's see how we can do that with an example in a stepwise process for example I wanted to add Say my algebraic expression a initially is 2x minus 5 plus 7x square and my algebraic expression say another algebraic expression b is 4 plus 3x minus 2x square. Now in this case I see that the like terms are not put one below the other. So we are going to rearrange and then add using the step 1 rule. So in this case, I first take A. Initially, I have this. Let me write them in higher order form. So first I write the highest degree which is 7x square. Then I write the next degree which is 2x. Then I write the next which is a constant. So let's maintain a chronological order where we start with the highest degree to the lowest degree. 2, 1, 0. The degree of this term is 2, the degree of this term is 1 and the degree of this term is 0. Similarly, I take the expression b where the highest degree of the term is negative 2x squared and the next highest degree is 3x. And the next highest is 4. So this is how I write them in a likewise term. So all the like terms are written one below the other and then only I can add. So once I add the two expressions, the left hand side becomes A plus B. And when I add the terms here, I use the step 2 rule which says that add the respective coefficients of like terms and write the respective exponent besides. So the respective coefficients in the first like terms are 7 and minus 2. So I simply add 7 and minus 2. 2 and then I put the coefficient the respective coefficient whatever is besides that then I add the respective coefficients for this and then write the x then I add the respective coefficients of the last terms and then we get this. Then finally what I do is I re-simplify this by adding further where 7 plus minus 2 is 7 minus 2 which is 5 x square and then plus I get 2 plus 3 which is 5 x so 5 x and this will be negative 5 plus 4 which is negative 1. So overall this is what I get for the sum of two algebraic expressions. So the expression a and expression b when I add them I get 5x squared plus 5x minus 1 using two important steps which we have discussed here. Now this can be applied for any problem. So any problem we take into consideration we should be able to add the two expressions based on the two simple process of rules. Next let's see one more example which is Quite different and let's see how we can do that in the same step 1 step 2 process so let's take another example where I take the expression P as 4x cubed minus 7x plus 5x square plus 3 is one of the expression which I have taken and let me take another expression say q is 2x minus 5x cube plus 9. Now these are the two different expressions but one thing what I identify between the two expressions is This does not form a chronological order where I have the degree here as 3, then degree 2, then degree 1 and degree 0. But I have degree 3. Degree 2 is missing, but degree 1 is there and the degree 0 term is there. So even these type of terms can be added with the same rules what we have taken in step 1 and step 2. So let us see how we are going to do that without getting confused in doing this problem. So when I take the expression p, I write in the chronological order as I told you the first in the example in the previous example always maintain a chronological order of writing the term. So first I take the highest degree term which is 4x cube then the next highest degree term is 5x square then the next highest degree is negative 7x and the next is degree 0 which is 3. And I maintain the same chronological order for the next algebraic expression, writing all like terms together, column wise. So for the second expression q, the highest degree term is minus phi xq. Then the next highest degree is degree 2, which I don't find. That term is missing out here. So I just put it blank. I don't write anything below x squared. Then I take the next highest degree which is degree 1 where I write this as plus 2x plus 9. So the value below 5x square is kept vacant because I maintain the like term column wise rule. So in this case without disturbing this term we just add them in this manner. So when I add I get p plus q is p plus q is equal to p minus 2x square. 4 minus 5 of x cube plus, now because I don't have any term, this is the term which comes directly falling in here. So this will be directly 5x square without any term being here. Then comes minus 7 plus 2 times of x. plus 3 plus 9. So we are adding the like coefficients together using the step 2 process. Now this would be nothing but 4 plus minus 5 is negative 1 of x cube. 5x square is as it is directly without any further simplification. This is negative 5x plus 12 which is the final answer for the sum of two given expressions. So this is how we do for any problem where we have some degrees missing. So we just keep them vacant. So whenever you find any degree missing out here, we just keep that vacant. That whole of the column is kept vacant and jump to the next degree and then write in this process. Now let's go to the next example. Now the two examples what we have seen have been with only two algebraic expressions which we have added in both of the cases. Now what happens when we have more than two algebraic expressions? Can we add three algebraic expressions together? Can we add four algebraic expressions together? Yes. So let's see one problem where you have more than two algebraic expressions and you need to add them. Say for example I have a. S2x minus 3 plus 4x squared and b is x minus 2 plus 5x cubed and my c is 1 plus x cubed. Now I wanted to add these three algebraic expressions using the step 1 and step 2 process together. So the three algebraic expressions are equally added in the same manner as we have added the two algebraic expressions. The rules do not change. They are the same. So the first rule says we are going to maintain the chronological order. So I take the highest degree which is x cube out of all the three expressions. So nothing is I don't have any term containing the of degree 3. So I just keep it. vacant. Then I write the next highest degree which is 4x square after the vacant place and the next highest degree is 2x and the next highest is negative 3. Same way for the next algebraic expression b the highest degree term is 5x cube which I write in the first. Of course this is vacant here because there's no x cube term in algebraic expression a. Next, the next degree is 4. The degree of 2 is missing. So I keep this vacant. Then I go to the next, which is x, and then minus 2. So this is how I maintain the chronological order. The x squared term missing. So this is kept vacant. Then I have my algebraic expression for c, which is 1 plus x cubed. So I write the like term x cubed below this. And there's one here. I don't have any term of x squared. x so I just keep it vacant. Now I have to add them so what I do is I just add the left hand side which is a plus b plus c then I add the right hand side which has these two terms with coefficients added respectively. Then, because I don't have any terms below this, this is the term which directly falls in here, which is 4x squared. And then I add the x terms, which can be added as 2 plus 1 times of x. Then I have all the constants in each of the expressions, so I have all the three terms in the form minus 3, minus 2, plus 1. Then finally, the simplification. each of the coefficients further gives me the sum of all three algebraic expressions. So this is 6x cubed plus 4x squared plus 3x minus 5 plus 1 is minus 4. So therefore the sum of all the three algebraic expressions given as ABC would be 6x cubed plus 4x squared plus 3x minus 4. So this is how we add. So we'll see subtraction of algebraic expressions the way we are seen adding of two algebraic expressions. It is quite same, the only difference being that when we add we put plus, when we subtract we put minus. So let's take one example directly and see how the two algebraic expressions can be subtracted, one from the other. So in case of subtraction, let me take the algebraic expression A. as 2x minus 1 plus x square and the algebraic expression b as 2x square plus 4x plus 3. Now I wanted to find a minus b. So whenever I subtract b from a, I write the algebraic expression a first and then the algebraic expression b. But unlike in addition of two algebraic expressions, the freedom is there in writing any algebraic expression initially. But when it comes to subtraction, I write the first algebraic expression a initially and then below that the algebraic expression b. The two steps are the same. What we have seen for the previous two examples, we write all the like terms together and then subtract instead of adding because here the binary operation is subtraction. So let's take the algebraic expression in chronological order where the highest degree is x squared, then the next degree is 2x and the next is 1. Now below this, I'm writing this so I take a as this and below this I write the algebraic expression b in chronological order as 2x square plus 4x plus 3 with all like terms together in column wise. Now the only additional rule I include in subtraction of two algebraic expressions is when I subtract all the signs of the second expression change. plus becomes minus and minus becomes plus. So when I have plus b it becomes minus, plus 2x square becomes minus 2x square, 4x becomes minus 4x and plus 3 becomes minus. So all plus signs become minus or any other minus sign you find here becomes plus. So using that we can really subtract. So subtraction of the two expressions is subtraction of each of the terms of the expressions, the like terms, subtraction of each of the like terms of the given expressions. So when I have these two expressions, it is nothing but subtraction of the coefficients of each of the. expressions. Then I have 2 plus minus 4 of x then I have negative 1 of negative 3. So this is how I get subtracting each of the expressions. Then this would be 1 minus 2 is minus 1 x squared and 2 minus 4 is minus 2 of x minus 1 minus 3 is minus 4. So my algebraic expression a minus b. finally obtained as negative x square negative 2x negative 4. So this is how I get the difference of two algebraic expressions or subtraction of two algebraic expressions. Yes. Next, if suppose I have a combination of addition and subtraction together If suppose I have, say, I wanted to find, if suppose I have the algebraic expression a as 2x squared minus 1 plus 3x, and algebraic b as 1 minus x cubed, and algebraic c as 2x minus 3. Yes. So in this case, I wanted to find the sum of two algebraic expressions minus the third expression. So this problem is a combination of both addition and subtraction. So we use the same process. We can do either by adding first a plus b and then subtracting that expression with c or you can do all together. But I suggest let's go step by step process because it's easy in simplifying in a step by step process. So when I have A plus B, so let me first find A plus B and then next I subtract C from that obtained. expression. So we all know that a plus b is nothing but addition of two expressions. So first let me write the expression a in chronological order plus 3x minus 1 and then let me write the expression b below that with the like terms minus x cube plus 1. Oh, I should be leaving some space here because the highest degree of these two expressions is 3. So, leaving a space, I get here minus x cubed plus 1 with these two places left vacant. Now, I am going to add these two. So, addition of these two expressions gives me a plus b on the left hand side. And when I add the like terms here, there is no like term in the... Expression A corresponding to expression B. So therefore directly I have this minus x cubed entering in the first term. Then I don't have anything below this so that comes directly as 2x squared plus 3x minus 1 plus 1 would be 0. So the constant vanishes in case of addition of these two expressions. So therefore, I get a plus b as negative x cubed plus 2x squared plus 3x. Now this is the expression I got in chronological order for a plus b. But when I want to find a plus b minus c, I just write c below this and I subtract these two expressions where indirectly I get this. So let's write the algebraic expression c in chronological order where I have 2x and I have negative 3. But I don't have any of these terms with degree 3 and degree 2 in c. So I just keep them vacant. Then in this case because I need to subtract, I have to subtract therefore I change all the signs of here minus, minus and minus becomes plus. All the signs become opposite. So in this case being minus, minus and plus. Therefore the left hand side reduces to a plus b minus c. And the right hand side, because there is no term below this, I directly have minus x cube entering in here. And then this is similar is the case with 2x squared, which enters in directly. Then I have 3x minus 2x, which is x. And this is plus 3, which is the answer for a plus b minus c. So like this we can add or subtract together or you can add all terms you can take four expressions and do addition and subtraction. There are many things which we can do out here. So this is how we obtain the addition and subtraction of algebraic expressions.

So there are totally many different types of algebraic expressions. To start with the first algebraic expression is a monomial. A monomial is an algebraic expression which has only one term. So for example, algebraic expression with only one term. For example, if I just take 3x, 5x squared.

and minus 7x or I take 11. So all these are monomials because they are just written with only one term. So same way if I just say binomial, it is an algebraic expression with two terms is called a binomial. So let's start with Monomial and the next is binomial and then comes trinomial and so on and so forth.

So binomial is an algebraic expression with two terms. So this is the definition for a binomial expression. So for example, if I just take... 2x minus 3, I identify here that there are two terms, one is 2x and one is negative 3. So the combination of two terms written in an expression form is called a binomial expression.

So an expression written with only one term is called a monomial expression. So I would like to take one more example where 5x squared minus 7 also is a binomial. If I have two unknown variables, say 2x plus 3x square is also a binomial, where the first term is 2x and the second term is 3x square. Let me take one interesting example where x square plus 3x minus 4x square.

Now here, I identify that This is not in the binomial form, but I can make it converted into binomial expression. Because these two terms can be further simplified to reduce to one particular term, so that you get two terms reduced after simplification. So this indirectly is a binomial expression in the way that, when I just take x square minus 4x square is minus 3x square, and plus 3x written out here. So the simplification of x square and negative 4x square gives me minus 3x square plus 3x fixed and written as the second term, which makes me understand that this is the first term and the second term. Because it has two terms in the expression, this is an example of a binomial expression.

Whereas if you take the unsimplified form, it doesn't form a binomial. expression. So, these are the various examples for binomial. Now, once monomial and binomial has been discussed, the third thing what is there in the definition is trinomial. And you all can guess what a trinomial is based on the definitions which we have taken there.

An algebraic expression with one term is monomial, with two terms is binomial, so with three terms will be a trinomial. So the definition is quite easy. Algebraic expression with three terms and even there are so many examples which we can take out here. If I take an example where 5 minus 3x plus 7x squared is a trinomial because it has three terms as it can be clearly seen.

The first term is 5, second term is negative 3x and the third term is 7x power 2. So these three terms makes this algebraic expression to be called as a trinomial or a trinomial expression. So there are many many examples which we can take in mathematics related to trinomial. Yes, so these are the three basic examples which are taken.

One is the monomial, binomial and trinomial. So there are many expressions where you can have more than three, you can have four, five, six. So generally any expression which has more than three terms is called a multinomial. So we need to, we need not, we cannot take the number of terms defined for any number of expressions where the terms are increased to a larger extent.

It is worth meaningful to take one term as monomial, two terms as binomial and three terms as trinomial. But when you have n number of terms, it is called a multinomial. So an algebraic expression with n terms with n greater than 3 is called a multinomial.

Where for n equal to 1, 2, 3 is already defined for monomial, binomial and trinomial. So let's see an example for a multinomial. For example, I take an algebraic expression 3 minus 4x plus 5x squared plus 7x cubed minus 9x power 4 minus 9x power 4 plus 12x power 5. Now this algebraic expression, let's see how many terms are there and we'll count by the process. 1, 2, 3, 4, 5, 6. So this is an algebraic expression with six terms which is called a multinomial with six terms.

So this is an example of a multinomial consisting of six terms. Now, based on all the various types of algebraic expressions which we have seen, starting with monomial, binomial, trinomial and multinomial, we will now see how we are going to add or subtract the different algebraic expressions using the like terms and unlike terms. So first, initially, we will see what exactly is a like term and what exactly is the unlike terms.

So when we discuss about like terms, the terms where the degree is same are called the like terms. So if the degree is same, then the terms are like terms. So when I say degree, the highest power of the unknown variable, say for example, I take 3x minus 5x square, 4x, 7x cube. So in this case, I find that the degree of this term is 1 because the power of x.

is 1 here. Now when I take here the degree of the second term is 2 because the power of the unknown variable that is x is 2. If I take here the degree of this term is 1 because x has its power as 1. When I take this term, its degree is 3 because of x cube. Now in this case, I identify that the first term and the third term are having the same degree.

So these two are said to be the like terms. So the first term and the third term have same degree 1. Therefore, according to the definition of like terms, 3x and 4x are called Like terms. So whenever we have any problem where we need to identify the like terms, we just search for the degree and if the degree is same, they are like terms and if the degree is not same, they are not the like terms.

So when we define the like terms, similar fashion we can define the unlike terms. If the degree is not same, then the terms are unlike terms. So here when I take 3x and minus 5x square these two are having two different degrees.

Therefore these two are unlike terms. These two are like terms. 3x and 7x cube are unlike terms because the degree of this is 1 and degree of this is 3. So let's take it as a different example definition and then see many examples of unlike terms.

So let's see the definition of unlike terms, the way we are seen for like terms. So unlike terms are the terms whose degrees are not same. So let's take the definition, the terms whose degree... are not same.

They are different. They are called unlike terms. Say for example, 5x and 7x cube.

I find that the degree of 5x is 1 and the degree of 7x cube is 3. So these two are with different degrees. So these two are called the unlike terms. So therefore they are different. unlike terms the reason being with two different degrees So after we understood the concept of like terms and unlike terms, the concept of like terms is very much useful when we add or subtract the two algebraic expressions or more than two algebraic expressions. Because when we add two algebraic expressions, we write all the like terms in column wise.

And let's see how we are going to do that. So the topic here is adding of. Algebraic expressions.

So before we proceed with how we are going to add, we are going to take certain rules into consideration. Step 1. We write all like terms together. Write all like terms. of each algebraic expression one below the other. So we write all like terms one below the other how many our algebraic expressions we have that is the first step we are going to follow.

Step 2 After we write all the like terms one by one, one below the other, we add the coefficients of each term individually and then write the power of x besides. Add the respective coefficients of like terms and write the exponent whichever is the degree respectively so after that we obtain the sum of two or more than two algebraic expressions so these are the two important steps we need to follow before we proceed with adding so let's see how we can do that with an example in a stepwise process for example I wanted to add Say my algebraic expression a initially is 2x minus 5 plus 7x square and my algebraic expression say another algebraic expression b is 4 plus 3x minus 2x square. Now in this case I see that the like terms are not put one below the other. So we are going to rearrange and then add using the step 1 rule. So in this case, I first take A.

Initially, I have this. Let me write them in higher order form. So first I write the highest degree which is 7x square. Then I write the next degree which is 2x. Then I write the next which is a constant.

So let's maintain a chronological order where we start with the highest degree to the lowest degree. 2, 1, 0. The degree of this term is 2, the degree of this term is 1 and the degree of this term is 0. Similarly, I take the expression b where the highest degree of the term is negative 2x squared and the next highest degree is 3x. And the next highest is 4. So this is how I write them in a likewise term. So all the like terms are written one below the other and then only I can add. So once I add the two expressions, the left hand side becomes A plus B.

And when I add the terms here, I use the step 2 rule which says that add the respective coefficients of like terms and write the respective exponent besides. So the respective coefficients in the first like terms are 7 and minus 2. So I simply add 7 and minus 2. 2 and then I put the coefficient the respective coefficient whatever is besides that then I add the respective coefficients for this and then write the x then I add the respective coefficients of the last terms and then we get this. Then finally what I do is I re-simplify this by adding further where 7 plus minus 2 is 7 minus 2 which is 5 x square and then plus I get 2 plus 3 which is 5 x so 5 x and this will be negative 5 plus 4 which is negative 1. So overall this is what I get for the sum of two algebraic expressions.

So the expression a and expression b when I add them I get 5x squared plus 5x minus 1 using two important steps which we have discussed here. Now this can be applied for any problem. So any problem we take into consideration we should be able to add the two expressions based on the two simple process of rules. Next let's see one more example which is Quite different and let's see how we can do that in the same step 1 step 2 process so let's take another example where I take the expression P as 4x cubed minus 7x plus 5x square plus 3 is one of the expression which I have taken and let me take another expression say q is 2x minus 5x cube plus 9. Now these are the two different expressions but one thing what I identify between the two expressions is This does not form a chronological order where I have the degree here as 3, then degree 2, then degree 1 and degree 0. But I have degree 3. Degree 2 is missing, but degree 1 is there and the degree 0 term is there. So even these type of terms can be added with the same rules what we have taken in step 1 and step 2. So let us see how we are going to do that without getting confused in doing this problem.

So when I take the expression p, I write in the chronological order as I told you the first in the example in the previous example always maintain a chronological order of writing the term. So first I take the highest degree term which is 4x cube then the next highest degree term is 5x square then the next highest degree is negative 7x and the next is degree 0 which is 3. And I maintain the same chronological order for the next algebraic expression, writing all like terms together, column wise. So for the second expression q, the highest degree term is minus phi xq. Then the next highest degree is degree 2, which I don't find.

That term is missing out here. So I just put it blank. I don't write anything below x squared.

Then I take the next highest degree which is degree 1 where I write this as plus 2x plus 9. So the value below 5x square is kept vacant because I maintain the like term column wise rule. So in this case without disturbing this term we just add them in this manner. So when I add I get p plus q is p plus q is equal to p minus 2x square. 4 minus 5 of x cube plus, now because I don't have any term, this is the term which comes directly falling in here. So this will be directly 5x square without any term being here.

Then comes minus 7 plus 2 times of x. plus 3 plus 9. So we are adding the like coefficients together using the step 2 process. Now this would be nothing but 4 plus minus 5 is negative 1 of x cube.

5x square is as it is directly without any further simplification. This is negative 5x plus 12 which is the final answer for the sum of two given expressions. So this is how we do for any problem where we have some degrees missing.

So we just keep them vacant. So whenever you find any degree missing out here, we just keep that vacant. That whole of the column is kept vacant and jump to the next degree and then write in this process. Now let's go to the next example.

Now the two examples what we have seen have been with only two algebraic expressions which we have added in both of the cases. Now what happens when we have more than two algebraic expressions? Can we add three algebraic expressions together? Can we add four algebraic expressions together?

Yes. So let's see one problem where you have more than two algebraic expressions and you need to add them. Say for example I have a. S2x minus 3 plus 4x squared and b is x minus 2 plus 5x cubed and my c is 1 plus x cubed.

Now I wanted to add these three algebraic expressions using the step 1 and step 2 process together. So the three algebraic expressions are equally added in the same manner as we have added the two algebraic expressions. The rules do not change. They are the same.

So the first rule says we are going to maintain the chronological order. So I take the highest degree which is x cube out of all the three expressions. So nothing is I don't have any term containing the of degree 3. So I just keep it.

vacant. Then I write the next highest degree which is 4x square after the vacant place and the next highest degree is 2x and the next highest is negative 3. Same way for the next algebraic expression b the highest degree term is 5x cube which I write in the first. Of course this is vacant here because there's no x cube term in algebraic expression a. Next, the next degree is 4. The degree of 2 is missing. So I keep this vacant.

Then I go to the next, which is x, and then minus 2. So this is how I maintain the chronological order. The x squared term missing. So this is kept vacant.

Then I have my algebraic expression for c, which is 1 plus x cubed. So I write the like term x cubed below this. And there's one here.

I don't have any term of x squared. x so I just keep it vacant. Now I have to add them so what I do is I just add the left hand side which is a plus b plus c then I add the right hand side which has these two terms with coefficients added respectively.

Then, because I don't have any terms below this, this is the term which directly falls in here, which is 4x squared. And then I add the x terms, which can be added as 2 plus 1 times of x. Then I have all the constants in each of the expressions, so I have all the three terms in the form minus 3, minus 2, plus 1. Then finally, the simplification. each of the coefficients further gives me the sum of all three algebraic expressions.

So this is 6x cubed plus 4x squared plus 3x minus 5 plus 1 is minus 4. So therefore the sum of all the three algebraic expressions given as ABC would be 6x cubed plus 4x squared plus 3x minus 4. So this is how we add. So we'll see subtraction of algebraic expressions the way we are seen adding of two algebraic expressions. It is quite same, the only difference being that when we add we put plus, when we subtract we put minus.

So let's take one example directly and see how the two algebraic expressions can be subtracted, one from the other. So in case of subtraction, let me take the algebraic expression A. as 2x minus 1 plus x square and the algebraic expression b as 2x square plus 4x plus 3. Now I wanted to find a minus b.

So whenever I subtract b from a, I write the algebraic expression a first and then the algebraic expression b. But unlike in addition of two algebraic expressions, the freedom is there in writing any algebraic expression initially. But when it comes to subtraction, I write the first algebraic expression a initially and then below that the algebraic expression b.

The two steps are the same. What we have seen for the previous two examples, we write all the like terms together and then subtract instead of adding because here the binary operation is subtraction. So let's take the algebraic expression in chronological order where the highest degree is x squared, then the next degree is 2x and the next is 1. Now below this, I'm writing this so I take a as this and below this I write the algebraic expression b in chronological order as 2x square plus 4x plus 3 with all like terms together in column wise. Now the only additional rule I include in subtraction of two algebraic expressions is when I subtract all the signs of the second expression change.

plus becomes minus and minus becomes plus. So when I have plus b it becomes minus, plus 2x square becomes minus 2x square, 4x becomes minus 4x and plus 3 becomes minus. So all plus signs become minus or any other minus sign you find here becomes plus. So using that we can really subtract.

So subtraction of the two expressions is subtraction of each of the terms of the expressions, the like terms, subtraction of each of the like terms of the given expressions. So when I have these two expressions, it is nothing but subtraction of the coefficients of each of the. expressions. Then I have 2 plus minus 4 of x then I have negative 1 of negative 3. So this is how I get subtracting each of the expressions. Then this would be 1 minus 2 is minus 1 x squared and 2 minus 4 is minus 2 of x minus 1 minus 3 is minus 4. So my algebraic expression a minus b.

finally obtained as negative x square negative 2x negative 4. So this is how I get the difference of two algebraic expressions or subtraction of two algebraic expressions. Yes. Next, if suppose I have a combination of addition and subtraction together If suppose I have, say, I wanted to find, if suppose I have the algebraic expression a as 2x squared minus 1 plus 3x, and algebraic b as 1 minus x cubed, and algebraic c as 2x minus 3. Yes. So in this case, I wanted to find the sum of two algebraic expressions minus the third expression. So this problem is a combination of both addition and subtraction.

So we use the same process. We can do either by adding first a plus b and then subtracting that expression with c or you can do all together. But I suggest let's go step by step process because it's easy in simplifying in a step by step process.

So when I have A plus B, so let me first find A plus B and then next I subtract C from that obtained. expression. So we all know that a plus b is nothing but addition of two expressions. So first let me write the expression a in chronological order plus 3x minus 1 and then let me write the expression b below that with the like terms minus x cube plus 1. Oh, I should be leaving some space here because the highest degree of these two expressions is 3. So, leaving a space, I get here minus x cubed plus 1 with these two places left vacant.

Now, I am going to add these two. So, addition of these two expressions gives me a plus b on the left hand side. And when I add the like terms here, there is no like term in the... Expression A corresponding to expression B.

So therefore directly I have this minus x cubed entering in the first term. Then I don't have anything below this so that comes directly as 2x squared plus 3x minus 1 plus 1 would be 0. So the constant vanishes in case of addition of these two expressions. So therefore, I get a plus b as negative x cubed plus 2x squared plus 3x. Now this is the expression I got in chronological order for a plus b. But when I want to find a plus b minus c, I just write c below this and I subtract these two expressions where indirectly I get this.

So let's write the algebraic expression c in chronological order where I have 2x and I have negative 3. But I don't have any of these terms with degree 3 and degree 2 in c. So I just keep them vacant. Then in this case because I need to subtract, I have to subtract therefore I change all the signs of here minus, minus and minus becomes plus. All the signs become opposite. So in this case being minus, minus and plus.

Therefore the left hand side reduces to a plus b minus c. And the right hand side, because there is no term below this, I directly have minus x cube entering in here. And then this is similar is the case with 2x squared, which enters in directly. Then I have 3x minus 2x, which is x. And this is plus 3, which is the answer for a plus b minus c.

So like this we can add or subtract together or you can add all terms you can take four expressions and do addition and subtraction. There are many things which we can do out here. So this is how we obtain the addition and subtraction of algebraic expressions.

Transcript for:Understanding Algebraic Expressions and Operations

Transcript for:
Understanding Algebraic Expressions and Operations