units in c + d . By (53) a + b must be taken c times, and d times, or the product required is ( a + b ) c + ( a + b ) d . But (52) ( a + b ) c is ac + bc , and ( a + b ) d is ad + bd ; whence the product required is ac + bc + ad + bd ; or,
(a + b)(c + d) = ac + bc + ad + bd.
By similar reasoning
(a - b)(c + d) is (a - b)c + (a - b)d; or,
(a - b)(c + d) = ac - bc + ad - bd.
To multiply a-b by c-d, first take a-b c times, which gives ac-bc. This is not correct; for in taking it c times instead of c-d times, we have taken it d times too many; or have made a result which is (a-b)d too great. The real result is therefore ac-bc-(a -b)d. But (a-b)d is ad- bd, and therefore
(a - b)(c - d) = ac - bc - ad - bd
= ac - bc - ad + bd (41)
From these three examples may be collected the following rule for the multiplication of algebraic quantities: Multiply each term of the multiplicand by each term of the multiplier; when the two terms have both + or both-before them, put + before their product; when one has + and the other-, put-before their product. In using the first terms, which have no sign, apply the rule as if they had the sign +.
For example, ( a + b )( a + b ) gives aa + ab + ab + bb . But ab + ab is 2 ab ; hence the square of a + b is