 Hello and welcome to the session. In this session first we will discuss about algebraic expressions. We already know that expressions are formed from variables and constants. Let's consider the expression 2xy plus 3. This is an algebraic expression formed from variables and constants. The variables in this case are x and y and the constants are 2 and 3. Then you also know that value of an expression changes with the value chosen for the variables it contains. So for different values of x and y the value of this expression also changes accordingly. Now let's discuss about terms, factors and coefficients. We know that terms are added. To form expressions, let's consider the expression 2x plus 3. This is an algebraic expression in which we have two terms 2x and 3. These terms, the answers can be formed as a product of factors like the term 2x is formed by 2 into x. So 2 and x are the factors of the term 2x and the term 3 is made up of just one factor which is 3 itself. We have the numerical factor of a term is called its coefficient like the term 2x in the given expression has coefficient 2. So we say coefficient of the term 2x is 2. Now we shall discuss monomials, binomials and polynomials. If an expression contains only one term then we say that it is a monomial and if an expression contains two terms then we call it a binomial. If an expression contains three terms we call it trinomial. In general we can say that an expression containing one or more terms with nonzero coefficient and with variables having nonnegative exponents is called a polynomial. So a polynomial may contain any number of terms one or more than one. Consider the expression 3x square. Now since this expression contains one term so we say this is a monomial. Next we consider the expression 2 plus 2y. This expression contains two terms so we say this is binomial. Then we have 3x plus 4y plus 6. This expression contains three terms so this is a trinomial. Now the expression a plus b plus c plus d has four terms in it and so we can say that it is a polynomial. Next we discuss like and unlike terms. Like terms are formed from the same variables and the powers of these variables are the same too but the coefficients of like terms need not be same. Like for example 2x minus 7x these are all like terms. Now in the unlike terms the variables need not be same and the powers of the variables also may not be same. Like for example if you have 3x and 4y these are the unlike terms. Since the variables are different. Next we discuss addition and subtraction of algebraic expressions. While adding or subtracting the polynomials first we have to look for the like terms and add them or subtract them as required then we will handle the unlike terms in the given polynomials. Consider a polynomial 2x plus 3y plus 5z. We need to add this polynomial to the polynomial 4y plus 2x plus 3z. What we do is we write both the polynomials that is this is one polynomial and this is the other polynomial. In separate rows first we write this 2x plus 3y plus 5z and below this we write the next polynomial in such a way that we get the like terms one below the other. So we write this as 2x plus 4y plus 3z and we need to add both these polynomials. First let's consider this first column in which we have 2x plus 2x this gives us 4x. So we write 4x below this. Now then we have in the second column 3y plus 4y which gives us 7y since 3 plus 4 is 7. So we write here plus 7y. Let's consider this third column which has 5z plus 3z. This is equal to 8z since 5 plus 3 is 8 so this is plus 8z. Hence on adding the given 2 polynomials we get 4x plus 7y plus 8z. This is the answer. In the same way we can subtract the given 2 polynomials also. Now we discuss multiplication of algebraic expressions. First let's see how we multiply a monomial by a monomial. Let's multiply 2 monomials 2x and 5y that is 2x into 5y would give us 4x plus 7y plus 7y a monomial only because when we multiply 2 or more monomials we get a monomial as a result. So to get the coefficient of the product of these 2 monomials we multiply the coefficients of both these monomials that is we have 2 into 5 which gives us 10. So this is equal to 10 and then we multiply the algebraic factor of 1 monomial by the algebraic factor of the other monomial to get the algebraic factor of the product that is we have x into y which gives us xy so we write here xy. So 2x multiplied by 5y gives us 10xy as the product. In the same way we can multiply 3 monomials. Let's multiply 2x with 5y and 3z that is we need to find the product of 2x, 5y and 3z for this again we need to find the coefficient of the product which would be equal to the coefficient of the 3 monomials multiplied together that is 2 into 5 into 3 which gives us 30. So 30 is the coefficient of the product of these 3 monomials then the product of the algebraic factors of these 3 monomials is x into y into z that is xyz we write xyz here so 30xyz is the product of the given 3 monomials. So in this way we can find the product of any given number of monomials and the resulting product would be a monomial itself. Now let's see how we multiply monomial by a binomial consider the monomial 2x that needs to be multiplied by the binomial 3x plus 2 that is we need to find the product of the monomial 2x and the binomial 3x plus 2 in this case every term of this binomial is multiplied by the given monomial that is this is equal to 2x into 3x plus 2x into 2 that is we need to find the product of the monomial 2. Now we already know how we multiply 2 monomials so this would give us 6x square plus 2x into 2 is 4x. So this is the product of given monomial and binomial. Next let's see how we multiply monomial by a trinomial consider the monomial 2x we multiply this monomial 2x and the trinomial 2x plus 3y plus 2. So we need to find 2x into 2x plus 3y plus 2 in this case also we multiply each term of the trinomial by the given monomial. So this would be equal to 2x into 2x plus 2x into 3y plus 2x into 2 this gives us 4x square plus 6xy plus 4x. So this is the product of the given trinomial and monomial. As we seen both these cases that is well multiplying the monomial by binomial and monomial by trinomial we have used the distributive law that is a multiplied by b plus c is equal to a multiplied by b plus a multiplied by c. In case of trinnovials also we have use the distributive law that is a multiplied by b plus c plus d which gives us a multiplied by b plus a multiplied by c plus a multiplied by d. Now let's see how we multiply a polynomial by a polynomial. First let's see how we multiply a binomial by a binomial, consider a binomial 2x plus 3y to be multiplied by 2x plus 5z. That is we need to find the product 2x plus 3y multiplied by 2x plus 5z. In this case we multiply every term in one binomial with every other term in the other binomial. Like this would be equal to the term 2x of this binomial is multiplied by 2x plus 5z plus 3y into 2x plus 5z. Now we follow the distributive law. So this gives us 2x into 2x plus 2x into 5z plus 3y into 2x plus 3y into 5z. This is equal to 4x square plus 10xz plus 6xy plus 15yz. Now in this case as you can see there are no like terms but in case if we have like terms in the product then we combine those like terms. Now let's see how we multiply a binomial by a trinomial, consider the binomial 2x plus 4y. We need to multiply this and the trinomial 2x plus 3y plus 5z. That is we need to find the product of 2x plus 4y and 2x plus 3y plus 5z. Each term of the trinomial is multiplied by each term of the binomial. That is this is equal to 2x into 2x plus 3y plus 5z plus 4y into 2x plus 3y plus 5z. Now following the distributive law we get 2x into 2x plus 2x into 3y plus 2x into 5z plus 4y into 2x plus 4y into 3y plus 4y into 5z. And this is equal to 4x square plus 6xy plus 10xz plus 8xy plus 12y square plus 20yz. Now in this product let's look out for the like terms and combine them. So as you can see 6xy and 8xy are the like terms. When they get combined we get 14xy. So the resulting product would be 4x square plus 14xy plus 10xz plus 12y square plus 20yz. Now let's discuss what is an identity. Basically identity is an equality which is true for all values of the variables in the equality. On the other hand we say that an equation is true for only certain values of the variable in it. That is we say that an equation is not an identity. Like if you consider a plus 3 multiplied by a plus 4 equal to a square plus 7a plus 12 for a equal to minus 3 and minus 4 we get LHS would be equal to the RHS. So this would be an identity. Since it is true for all values of the variable. Now we shall discuss some standard identities. Like the first one we have a plus b the whole square is equal to a square plus 2ab plus b square. Then the second identity is a minus b the whole square is equal to a square minus 2ab plus b square. Then next one that we have is a plus b into a minus b is equal to a square minus b square. Now another useful identity is x plus a into x plus b is equal to x square plus a plus b into x plus a b. Let's find the product x plus 3 into x plus 3 by using any of the above identities. Now this could be written as x plus 3 the whole square. So for this as you can easily see we will use the first identity that is a plus b the whole square equal to a square plus 2ab plus b square. In this case we would put a equal to x and b equal to 3. So this would give us x square plus 2 into x into 3 plus 3 square that comes out to be equal to x square plus 6x plus 9. So we get x plus 3 into x plus 3 is equal to x square plus 6x plus 9. So this is how we can use different identities to find these squares and products of algebraic expressions. This completes the session. Hope you have understood the concept of algebraic expressions and identities.