 Hello and welcome to the session. In this session first we will discuss factors of algebraic expressions. First let's see how we write the factors of natural numbers. Consider the number 20. The factors of 20 can be written as 2 into 2 into 5. These 2 and 5 are the prime numbers. So we say that this is the prime factorization of the number 20. In the same way we can also express the algebraic expressions as the product of their factors. Consider the algebraic expression 8xy plus 2x. Now we have two terms in this expression 8xy and 2x. 8xy can be written as 2 into 2 into 2 into x into y and then this 2x can be written as 2 into x. This is the prime factorization of the term 8xy. Now as you can see that this cannot be factorized further and in 2x also this cannot be factorized further. So 2x and y are the irreducible factors of 8xy and 2nx are irreducible factors of 2x. Now let's define an irreducible factor. It is a factor which cannot be expressed further as a product of factors or we can say this is the irreducible form of the term 8xy and this is the irreducible form of the term 2x. Next we see what is factorization. When we factorize expression we write it as a product of factors and these factors may be numbers, algebraic values or algebraic expressions. If you consider the expression 5xy then you can see that this is already in the factor form and we can make out its factor by just reading them that is 5x and y are its factors. Consider the expression 3x plus 9. We cannot make out what are the factors for this expression so we need some method to factorize expressions of these kinds that is we need some method to find their factors. First we discuss method of common factors. Let's consider the expression 3x plus 9. Now in this method we have three simple steps to factorize a given expression. In the first step what we do is we write each term of the given expression as a product of irreducible factors so the expression that we've given is 3x plus 9. Now it has two terms 3x and 9. Now we write each term as a product of irreducible factors that is we can write 3x as 3 into x then we write 9 as 3 into 3. So we get 3x plus 9 is equal to 3 into x plus 3 into 3. Now let's see what is the next step. In the next step we find out the common factor in both these terms and separate that. Now as you can see that 3 is the common factor in both these terms then in the next step we combine the remaining factors in each term in accordance with the distributive law. So by distributive law we get that 3 into x plus 3 is equal to 3 into x plus 3 into 3. So we write 3x plus 9 as 3 into x plus 3 or we write 3x plus 9 as 3 into x plus 3. So the factors of the given expression 3x plus 9 are 3 and x plus 3 and these are the irreducible factors. So this is how we use the method of common factors to factorize a given expression. The next method that we discuss is factorization by regrouping terms. When all the terms in a given expression do not have a common factor then what we can do is we can group those terms in such a way that all the terms in each group have a common factor. Consider the expression 15xy plus 5y minus 6x minus 2. Now while factorizing this given expression by regrouping terms let's see if they have any common factor. Now considering all these terms you can see that there is no common factor in all these terms. So now we can think of grouping them. Notice that the first two terms have a common factor 5y. So 15xy plus 5y can be written as 5y into 3x plus 1. Now let's consider the other two terms minus 6x minus 2. The common factor in minus 6x minus 2 is minus 2. So we take minus 2 out and inside we are left with 3x plus 1. Now we can put both these together that is we have 15xy plus 5y minus 6x minus 2 equal to 5y into 3x plus 1 minus 2 into 3x plus 1. Now 3x plus 1 is a common factor in both these terms. So we have 3x plus 1 into 5y minus 2. So the given expression 15xy plus 5y minus 6x minus 2 is written as 3x plus 1 into 5y minus 2. So this is how we can factorize a given expression by regrouping its terms. When we are given an expression and we need to factorize that expression first we shall check out if all the terms of that expression have a common factor or not. If they have a common factor then that can be factorized by a common factor method and if they don't have a common factor then we will use the method of regrouping the terms. Now let's discuss factorization using identities. We are familiar with the identity a plus b the whole square is equal to a square plus 2ab plus b square then a minus b the whole square is equal to a square minus 2ab plus b square and also a plus b into a minus b which is equal to a square minus b square. Now we will use these identities to factorize a given expression. Consider the expression 25n square plus 30m plus 9 this can be written as 5m the whole square plus 2 into 5m into 3 since we can write 9 as 3 square so this is of the form a square plus 2ab plus b square where we have a is equal to 5m and b is equal to 3. Now we know that a square plus 2ab plus b square is equal to a plus b the whole square. So now we can say that 25m square plus 30m plus 9 is equal to a plus b the whole square and we put values of a and b in this term so we have 5m plus 3 the whole square. So this is the required factorization. Now we have one more identity that is x plus a into x plus b is equal to x square plus a plus b into x plus a b this identity would be used to factorize the expression with one variable in it and also the expressions which are not perfect squares. In general we say that for factorizing an algebraic expression of the type x square plus bx plus q we find two factors a and b such that in general we say that to factorize an algebraic expression of the type x square plus bx plus q we need to find two factors a and b of q that is of the constant term such that we have ab is equal to q and a plus b is equal to p that is sum of all the factors a and b is p which is the coefficient of x in the given expression and the product of the two factors a and b is q which is the constant term of the given expression. Consider the expression x square plus 8x plus 16 now here we have the constant term that is q is 16 and this is p that is 8 now to factorize this given expression we need to find two factors a and b such that the product is 16 that is q and the sum is p that is 8. Now we can write 16 as 4 into 4 and we can write 8 as 4 plus 4 so we say that a is 4 and b is 4 that is the two factors are 4 and 4 and therefore the given expression x square plus 8x plus 16 is written as x square plus 4x plus 4x plus 16 now you can group these terms and so we get x into x plus 4 plus 4 into x plus 4 now x plus 4 is common to both these terms so we get x plus 4 into x plus 4 or you can say x plus 4 the whole square is the required factorization of the expression x square plus 8x plus 16 this completes the session hope you have understood the factorization of algebraic expressions.