Case 1: Suppose we have two terms 8ab and 4ab. On dividing the first by the second we have 8ab/4ab = 2 or 4ab/8ab = (1/2) depending on whether we consider either 8ab or 4ab as the first term. Irrespective of the order of division, the quotient of two positive terms will be a positive term.
Case 2: What will be the quotient if you divide -9ac by -3ac. We will get -9ac/-3ac = 3. In case of -3ac/-9ac, we will get 1/3. As in case 1, irrespective of the order of division, the quotient will be a positive term.
(Note: Observe that -9ac/-3ac is same as -9ac x 1/-3ac; 1/-3ac being the reciprocal of -3ac.)
We took monomials only for the sake of understanding the underlying principles in a better manner. However, these principles can be applied equally well to binomials, trinomials and polynomials also. The next part deals with them.
Now, do you find any difference between -7abc - 3bc and -7abc + (-3bc) and -7abc - (-3bc). The terms -7abc - 3bc and -7abc + (-3bc) are one and the same, the third expression is certainly different. The idea of introducing the bracket is to convey that the quantity within the brackets ought to be treated as a single quantity. Therefore, whenever one removes the brackets the necessary changes ought to be made specially with respect to the signs of the terms. Otherwise, you may end up with a wrong solution. And since we are mainly concerned with the multiplication aspect whenever brackets come into picture we apply the same four rules we have seen while going through multiplication of like and unlike terms. The expression -7abc + (-3bc) on removal of the brackets will be -7abc + x -3bc, where ‘x' is the multiplicative sign. Since + x - gives us -, the expression will get simplified to -7abc - 3bc. The third expression -7abc - (-3bc) would be simplified to -7abc + 3bc on removal of the brackets. Regarding the change of signs whenever brackets are present, we make the following two important observations:
Whenever a ‘+' sign precedes a bracket, the brackets can be removed without changing the signs of the elements within the brackets.
Whenever a ‘-' sign precedes a bracket, they can be removed only by changing the signs of the elements within the brackets.
From these observations we also conclude that if you want to introduce a ‘+' sign and include some of the terms of an expression in brackets, you can do so without changing the sign of the terms irrespective of them being ‘+' or ‘-'. That is, -7abc - 3ab can be written as -7abc + (-3ab) or 8a + bc can be written as 8a + (+bc). But if you want to introduce a ‘-' sign, more attention has to be paid. Every time you want to introduce a ‘-' sign, the signs of the terms to be included in the bracket has to be changed. In other words, a term which has a ‘-' sign should be changed to ‘+' sign and the term which has a ‘+' sign should be changed to ‘-' sign. Now we look at few examples as to how the basic operations are conducted in case of binomials, trinomials and polynomials.