MULTIPLICATION IN ALGEBRA

 

       A variable may be multiplied by a number as well as by another variable. When we multiply a vaiable with a number we get a term which consists of number and the variable. Suppose 3 is a number and x is a variable, then their product will be 3 * x. It is customery not to write multiplication sign * in between number and variable or in between any two variables of the same term. Also we write in first place the number and to the right of the number the variable part of the term as 3x. A term consisting of number and variable part can again be multiplied by a number or a variable. A term can also be multiplied by antother term.

      When we multiply one variable with another variable, we write their product by putting first variable and thereafter second variable without any sign in between them. For example, if x and y are two variables which are to be multiplied, then their product is xy. This xy is treated different from x and y both.

      A variable may also be multiplied by itself. For example a variable x can be multiplied by x. In this case product will be xx but instead of writing product as such, we write it as x2. We read it as x squared or x to the power two. This x2 can further be multiplied by x. then their product will be x2x. Instead of writing it as such we write it as x3. We will read it as x cubed or x to the power three. If we multiply x3 by another x, we get x3x . This is written as x4 and read as x to the power four. Thus power of a variable tells us that how many times the variable has been multiplied to itself. So, x5 is x*x*x*x*x or xxxxx; y3 is y*y*y or yyy.

       A term may be positive or negative. Before a positive term, if it is alone or if it is first term of an expression, + sign is not written before it. Contrary to this, a negative term everywhere carries a - sign before it.

       A term may consists, (i) + or - sign, (ii) a number as its coefficient; and (iii) a variable part. If a term does not have + or - sign, its sign is +. Also if term does not have a number as its coefficient, then its cofficient is 1. [we cannot say that if term does not have a number as cofficient then its cofficient is 0, because term =coefficient * variable part and if coefficient is 0, then product of coefficient and variable part will be 0. Then term will become 0 which will not be correct unless variable part is also 0]

      We have read rules of powers or indices. We can recall them here as under:
Terms
Product of terms
x3 & x5 x8       The rule is that when we multiply two terms of same variable having different powers then in the product power of variable is the sum of powers. Here one x has power of 3 and other x has power of 5. In the product x8, power of variable x is sum of 3 & 5 which is 8. We can check the result. x3 is x*x*x and x5 is x*x*x*x*x. Then multiplication of x3 & x5 will be the same as multiplication of x*x*x & x*x*x*x*x. But product of x*x*x * x*x*x*x*x is x*x*x*x*x*x*x*x which is x8.[we have used sign * for clarity only. In practice sign * is not written]
(x3)5

x15       Here variable x has power of 3 and variable with power of 3 has again power of 5. Power simply tells that how many times that thing has been multiplied. Thus x3 telss that x hax been multiplied three times that is x3 is x x x. Therefore, we can write (x3)5 as (x x x)5. Now(x x x)5 is 5 times multiplication of thing within bracket which is x x x. This will give xxx xxx xxx xxx xxx. This is nothing but x15.

      So when a power is raised to a variable which already has some power, the resultant power of variable is the product of both powers.

x1 x.       Any variable whose power is not written carries power of 1.
x0 1       Any variable to the power 0 = 1
      We have used these rules in cases of variables, these rules are also true in case of numbers.
MULTIPLICATION OF TWO OR MORE TERMS
      Suppose you are asked to muliply 2 , 3, 5 and 7 that is you have to find out 2*3*5*7. What you will do is that first you will multiply 2 with 3, you will get 6. Then you will multiply 6 by 5, you will get 30. Then you will multiply 30 with 7 then you will get 210. What We mean to say is that while multiplying several numbers, first you multiply first and second number to get product of two numbers. Then to this product of two numbers, you multiply it to thirs number in order to get product of three numbers. Again you multiply product of three number with fourth number, then you get product of four numbers. Algebric term may contain:
(i)
 + or - sign [where term has no sign, it carries + sign]
(ii) coefficient [where coefficien is not written, coefficient of term is 1]
(iii) variable part [variable part is the portion of a term which is left after removing sign and coefficien].
      While multiplying one algebric term with other algebric term, we multiply:
(i)
 sign with sign. The resultant sign gives sign of product of two terms.
(ii) coefficient with coefficient. The product of two numbers gives coefficient of product of two terms.
(iii)
 variable part of one term with variable part of other term. This gives us variable part of product of two terms.
and write the product = (product of signs) (product of coefficients)( product of variables).
      After putting the product in above form we simplify the new term get the final product.
Wxamples:
1. Find product of 3x2 and -5xy.
Product=(product of signs) (product of coefficients)( product of variables)
Here sign of 3x2 is + and sign of -5xy is -. Coefficient of first term is 3 and coefficient of second term is 5 and variable part of first term is x2 and variable part of second term is xy. Hence-
Product of given two terms=(+ * -)(3*5)(x2* xy)
Or product of given terms =-15 x2*xy
We know that x2 * x =x3 [because x2 = x*x. Therefore, x2*x = x*x*x* = x3]
Therefore, product of given terms is -15 x3y
2. Multiply -13 x3y2z4 by -7xy.
Product=(product of signs) (product of coefficients)( product of variables)

Therefore, product of given terms= (- *-) (13*7)(x3y2z4 *xy)

Or product of given terms =+ 91 x4y3z4 [here powers of x and y have been added separately]
Or Product =91 x4y3z4
3. Find product of three terms (i) -2x2y (ii) -5xyz and (iii) -17x3y5z2.
Product=(product of signs) (product of coefficients)( product of variables)
Therefore, product of given terms = (-*-*-)(2*5*17)(x2y * xyz *x3y5z2)
Or product of given terms= -170 x5y7z3 [Here product of first two - signs have given resultantsign of + and on multiplying resultant sign + with third - sign, finally it has given - sign. While simplifying variable parts we have added powers of x, y and z separately.]
 
4. Multiply 3x5y2 by x-2y
Product=(product of signs) (product of coefficients)( product of variables)
Therefore, product of given terms=(+*+)(3*1)(x5y2 * x-2y)
Or product of given terms=3x3y3 [coefficient of term x-2y is 1 and sum of powers of x is (5-2)]
 
MULTIPLICATION OF TWO EXPRESSIONS:
       Where both expressions have only single term, we can find out product of those expressions in the manner used for finding out product of two terms. But where either both expressions consists of more than one term or one expression consists of one term and other expression consists of more than one term, we apply a little different procedure.
Look at the following expression of numbers:
        (5-3) * (2+3+4)
        We can solve or simplify above expression in two ways. We can solve figures within each bracket and then can multiply resultant figures as under:
        2 * 9
Which gives us 18.        
We can also solve above expression of numbers by multiplying each term first by 3 and then by -3 and put all terms in a row as under:
5*(2+3+4) -3*(2+3+4)
Or   10 + 15 + 20 -6 -9 -12  
Which gives us    45-27
    Which is 18.
We see that in both ways, we get the same result.
        In Algebra, we do not know values of variables and hence expressions within two brackets cannot be solved separately to give a single variable. Therefore, we adopt second method given above. Let there be two expressions x+y and p+q+r. For finding out product of these two expressions, we proceed as under:
Product of two expressions= (first expression)(second expression)
We put the expressions in brackets as under:
Product of given expressions= (x+y)(p+q+r)
Now we take first variable of first expression with it's sign and to its right we put second expression within bracket. Again we take next term of first expression with it's sign and put it in the same row onwards and immediately their after we put second expression in bracket. This process continue till all terms of first expression are exhauasted. Applying this method, we get:
Product =x(p+q+r) +y(p+q+r)
Now for removing brackets, we multiply each term of first bracket by x and each term of first bracket by y. Then we get:
Product = xp +xq + xr + yp +yq +yr
Finally, we simplify the expression by adding or subtracting similar terms, if any. Here we see that variable parts of all terms are different from each other. Therefore, no two or more terms can be converted into a single term. Hence expression cannot further be simplified.
 
Example: Multiply (x2+y -z) by (x-z2)
Product =(fist term)(second term)
Hence, product of given terms=(x2+y -z)(x-z2)
Or product of given terms = x2(x-z2) + y(x-z2)-z(x-z2)
Or product=x2*x -x2*z2 +y*x -y*z2 -z*x + z*z2
Or product=x3 -x2z2 +yx -yz2 -zx + z3
We can write it as x3 -x2z2 +xy -yz2 -xz + z3
We have changed yx into xy and zx into xz.If two numbers or variables are multiplied, we can put any of them first that is we can reverse order of putting them together. Result is not affected. We know that product of 2 & 3 is the same as product of 3 & 2.Similarly, product of x & y is the same as product of y & x. Hence xy =yx.
WHILE MULTIPLYING TWO EXPRESSIONS, WE CAN PUT ANY EXPRESSION FIRST.
Example: Multiply x+y+z by x2 + y
While multiplying,we can put above expressions either as
Product = (x+y+z)(x2 + y)
Or as
Product =(x2 + y)(x+y+z)
Therefore, while multiplying two expressions, any expression can be taken as first expression. Normally, each term of expression having lesser number of terms is multiplied to expression having large number of terms.
 
MULTIPLICATION OF MORE THAN TWO EXPRESSIONS:
We put all expressions with small brackets in a row without any sign in between them. First we multiply first and second expression and we put their product in a bracket and rest of expression we put as such. Next, we multiply expression product with third expression to get product of three expressions. Similarly, we put product of three expressions in a bracket and multiply this by fourth expression. We get product of four expressions. The same process can be adopted for any number of expressions.
Example: Find product of (i) x+y (ii) x-y (iii) x-z and (iv) y-z.
Solution:
Product of four expressions = (first expression)(first expression)(first expression)(first expression)
Therefoe, product of given expressions=(x+y)(x-y)(x-z)(y-z)
Or Product={x(x-y) +y(x-y)}(x-z)(y-z)
Or product ={x2 -xy +yx-y2}(x-z)(y-z)
Or product = (x2 -y2)(x-z)(y-z) [since xy =yx and one being positive and other negative, their sum is 0]
Now we multiply product of first &second expression {which is (x2 -y2)} with third expression:
Therefore, product ={x2(x-z) -y2(x-z)}(y-z)
Or product = {x2x - x2z -y2-y2z}(y-z)
Or product=(x3-x2z -y2-y2z) (y-z)
Or product= (y-z)(x3-x2z -y2-y2z)
Or product =y(x3-x2z -y2-y2z) -z(x3-x2z -y2-y2z)
Or product =yx3-yx2z -y*y2-y*y2z -zx3-z*x2z -z*y2-z*y2z
Or product =x3y-x2yz-y3-y3z-x3z-x2z2 -y2z- y2z2 [re-organizing terms in alphabetical order and adding powers of same variable in the same term]
Variable parts of any two terms are not similar and hence no further simplification can be made.
 

 

 

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